{ "version" : "https://jsonfeed.org/version/1.1", "title" : "Aan Triono", "description": "Learn and Create — Sharing knowledge for the common good", "home_page_url" : "https://www.aantriono.com/", "feed_url" : "https://www.aantriono.com/en/index.feed", "icon" : "https://www.aantriono.com/apple-touch-icon.png", "favicon" : "https://www.aantriono.com/favicon.ico", "author" : { "name" : "Aan Triono", "url": "https://www.aantriono.com", "avatar": "https://www.aantriono.com/images/aan1.png" }, "items" : [ { "title" : "3D Spherical Polar Plots and Spherical Harmonics Visualization with tikz-3dplot", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/spherical-polar-pots-with-3dplot/", "url" : "https://www.aantriono.com/en/spherical-polar-pots-with-3dplot/", "summary": "This article explores how to visualize 3D mathematical functions in spherical polar coordinates and Spherical Harmonics using the tikz-3dplot package in LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores how to visualize 3D mathematical functions in spherical polar coordinates and Spherical Harmonics using the \u003ccode\u003etikz-3dplot\u003c/code\u003e package in LaTeX.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn quantum mechanics, geophysics, and mathematical physics, \u003cstrong\u003eSpherical Harmonics\u003c/strong\u003e \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmi\u003eY\u003c/mi\u003e\u003cmi\u003el\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msubsup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eY_l^m(\\theta, \\phi)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0331em;vertical-align:-0.2831em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eY\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6644em;\"\u003e\u003cspan style=\"top:-2.4169em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\"\u003el\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2831em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e represent a system of orthogonal functions that describe the angular dependence of wave equations in three-dimensional space. One of their most famous applications is representing atomic \u003cstrong\u003eelectron orbitals\u003c/strong\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ef\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003es, p, d, f\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) in hydrogen-like atoms.\u003c/p\u003e\n\u003cp\u003eRendering accurate 3D plots showing both the magnitude and phase of spherical functions directly inside LaTeX was once a challenging task. However, \u003cstrong\u003eJeff Hein\u003c/strong\u003e authored an extension for the \u003cstrong\u003e\u003ccode\u003etikz-3dplot\u003c/code\u003e\u003c/strong\u003e package that enables native 3D spherical polar surface plotting in TikZ.\u003c/p\u003e\n\u003cp\u003eIn these visualizations, the 3D geometric shape is defined by the function magnitude \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er(\\theta, \\phi)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, while parametric color shading (hue) conveys the complex phase angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\phi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code by Jeff Hein to visualize the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eY\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eY_{2,1}(\\theta, \\phi)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eY\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct mtight\"\u003e,\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e spherical harmonic function along with parameters for other quantum orbitals:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n 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CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath 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26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% harmonics.tex: produces spherical harmonic plots using the 3dplot package\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Jeff Hein\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminimal\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% for TikZ graphics\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz-3dplot\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% for 3dplot functionality\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[active,tightpage]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\PreviewEnvironment\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\\PreviewBorder\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Define 3D viewing angle (camera orientation: theta=70, phi=135)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tdplotsetmaincoords\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e70\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e135\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[line join=bevel,tdplot\u003cspan class=\"nb\"\u003e_\u003c/span\u003emain\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecoords, fill opacity=.7]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Spherical surface plot for L=2, M_L=1\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\tdplotsphericalsurfaceplot\u003c/span\u003e\u003cspan class=\"na\"\u003e[parametricfill]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e72\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e36\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(15/2)*sin(\u003cspan class=\"k\"\u003e\\tdplottheta\u003c/span\u003e)*cos(\u003cspan class=\"k\"\u003e\\tdplottheta\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tdplotphi\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=black,thick,-\u0026gt;]\u003c/span\u003e (0,0,0) -- (2,0,0) node[anchor=north east]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ex\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=black,thick,-\u0026gt;]\u003c/span\u003e (0,0,0) -- (0,2,0) node[anchor=north west]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ey\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=black,thick,-\u0026gt;]\u003c/span\u003e (0,0,0) -- (0,0,2) node[anchor=south]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ez\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"output-produced\" class=\"headerLink\"\u003e\n \u003ca href=\"#output-produced\" class=\"header-mark\" aria-label=\"Direct link to section: Output Produced\"\u003e\u003c/a\u003eOutput Produced\u003c/h2\u003e\u003cp\u003eCompiling the code above generates a beautiful 3D surface plot of the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eY\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eY_{2,1}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eY\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct mtight\"\u003e,\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e orbital:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/spherical-polar-pots-with-3dplot/spherical-polar-plots.webp\" title=\"Spherical Polar Plot 3D\" data-thumbnail=\"/spherical-polar-pots-with-3dplot/spherical-polar-plots.webp\" data-sub-html=\"\u003ch2\u003eCompiled 3D Spherical Harmonics Output\u003c/h2\u003e\u003cp\u003eSpherical Polar Plot 3D\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/spherical-polar-pots-with-3dplot/spherical-polar-plots.webp' srcset=\"/spherical-polar-pots-with-3dplot/spherical-polar-plots_hu_38df6656c2193ec.webp 800w, /spherical-polar-pots-with-3dplot/spherical-polar-plots_hu_bd9d1a591183a29d.webp 1200w, /spherical-polar-pots-with-3dplot/spherical-polar-plots_hu_9bca904ee6980862.webp 1600w\" alt=\"Spherical Polar Plot 3D\" height=\"311\" width=\"198\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCompiled 3D Spherical Harmonics Output\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-analysis--tikz-3dplot-components\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-analysis--tikz-3dplot-components\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Analysis \u0026amp; tikz-3dplot Components\"\u003e\u003c/a\u003eDetailed Code Analysis \u0026amp; \u003ccode\u003etikz-3dplot\u003c/code\u003e Components\u003c/h2\u003e\u003ch3 id=\"1-setting-up-3d-view-orientation\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-setting-up-3d-view-orientation\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Setting Up 3D View Orientation\"\u003e\u003c/a\u003e1. Setting Up 3D View Orientation\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tdplotsetmaincoords\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e70\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e135\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis command specifies the elevation angle (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e70\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\theta = 70^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) and azimuth angle (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e135\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\phi = 135^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) for the 3D perspective projection. Option \u003ccode\u003etdplot_main_coords\u003c/code\u003e is then passed to \u003ccode\u003etikzpicture\u003c/code\u003e so all coordinates align with this camera view.\u003c/p\u003e\n\u003ch3 id=\"2-syntax-of-tdplotsphericalsurfaceplot\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-syntax-of-tdplotsphericalsurfaceplot\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Syntax of \\tdplotsphericalsurfaceplot\"\u003e\u003c/a\u003e2. Syntax of \u003ccode\u003e\\tdplotsphericalsurfaceplot\u003c/code\u003e\u003c/h3\u003e\u003cp\u003eThe primary command \u003ccode\u003e\\tdplotsphericalsurfaceplot\u003c/code\u003e accepts several key arguments:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tdplotsphericalsurfaceplot\u003c/span\u003e\u003cspan class=\"na\"\u003e[options]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003ephi\u003cspan class=\"nb\"\u003e}{\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003etheta\u003cspan class=\"nb\"\u003e}{\u003c/span\u003efunction\u003cspan class=\"nb\"\u003e_\u003c/span\u003er\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eline\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecolor\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ephase\u003cspan class=\"nb\"\u003e_\u003c/span\u003efunction\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ex\u003cspan class=\"nb\"\u003e_\u003c/span\u003eaxis\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ey\u003cspan class=\"nb\"\u003e_\u003c/span\u003eaxis\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ez\u003cspan class=\"nb\"\u003e_\u003c/span\u003eaxis\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e[parametricfill]\u003c/code\u003e\u003c/strong\u003e: Enables spectrum gradient coloring based on the phase angle variable \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\phi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e{72}{36}\u003c/code\u003e\u003c/strong\u003e: Defines the grid resolution (72 azimuthal steps for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\phi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and 36 polar steps for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e{sqrt(15/2)*sin(\\tdplottheta)*cos(\\tdplottheta)}\u003c/code\u003e\u003c/strong\u003e: The mathematical equation defining the radial distance \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er(\\theta, \\phi)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e{\\tdplotphi}\u003c/code\u003e\u003c/strong\u003e: The function specifying the parametric hue variation.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCartesian Axes \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x,y,z)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/strong\u003e: The final three arguments allow drawing coordinate axes directly inside the plot environment.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"gallery-of-other-spherical-harmonics-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#gallery-of-other-spherical-harmonics-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Gallery of Other Spherical Harmonics Equations\"\u003e\u003c/a\u003eGallery of Other Spherical Harmonics Equations\u003c/h2\u003e\u003cp\u003eYou can swap the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eϕ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er(\\theta, \\phi)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϕ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e function expression to display various atomic orbital shapes:\u003c/p\u003e\n\u003ch3 id=\"1--orbital-\" class=\"headerLink\"\u003e\n \u003ca href=\"#1--orbital-\" class=\"header-mark\" aria-label=\"Direct link to section: 1. sss-Orbital (L=0,ML=0L = 0, M_L = 0L=0,ML​=0)\"\u003e\u003c/a\u003e1. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003es\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-Orbital (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL = 0, M_L = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3283em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/h3\u003e\u003cp\u003ePerfect spherical symmetry with constant radius:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"2--orbital-\" class=\"headerLink\"\u003e\n \u003ca href=\"#2--orbital-\" class=\"header-mark\" aria-label=\"Direct link to section: 2. pzp_zpz​-Orbital (L=1,ML=0L = 1, M_L = 0L=1,ML​=0)\"\u003e\u003c/a\u003e2. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep_z\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-Orbital (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL = 1, M_L = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3283em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/h3\u003e\u003cp\u003eTwo vertical lobes oriented along the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ez\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC 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class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"3--orbital-\" class=\"headerLink\"\u003e\n \u003ca href=\"#3--orbital-\" class=\"header-mark\" aria-label=\"Direct link to section: 3. px,yp_{x,y}px,y​-Orbital (L=1,ML=±1L = 1, M_L = \\pm 1L=1,ML​=±1)\"\u003e\u003c/a\u003e3. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep_{x,y}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7167em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct mtight\"\u003e,\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-Orbital (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL = 1, M_L = \\pm 1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3283em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e±\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/h3\u003e\u003cp\u003eHorizontal lobes along the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003exy\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-plane:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 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0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(3/2)*sin(\u003cspan class=\"k\"\u003e\\tdplottheta\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tdplotphi\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"4--orbital-\" class=\"headerLink\"\u003e\n \u003ca href=\"#4--orbital-\" class=\"header-mark\" aria-label=\"Direct link to section: 4. dz2d_{z^2}dz2​-Orbital (L=2,ML=0L = 2, M_L = 0L=2,ML​=0)\"\u003e\u003c/a\u003e4. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed_{z^2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8721em;vertical-align:-0.1776em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3448em;\"\u003e\u003cspan style=\"top:-2.5224em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7463em;\"\u003e\u003cspan style=\"top:-2.786em;margin-right:0.0714em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1776em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-Orbital (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL = 2, M_L = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3283em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/h3\u003e\u003cp\u003eTwo primary lobes along the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ez\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis with an equatorial doughnut torus:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(5)/2*(3*cos(\u003cspan class=\"k\"\u003e\\tdplottheta\u003c/span\u003e)\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2 - 1)\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\tdplotphi\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"5--orbital-\" class=\"headerLink\"\u003e\n \u003ca href=\"#5--orbital-\" class=\"header-mark\" aria-label=\"Direct link to section: 5. fff-Orbital (L=3,ML=0L = 3, M_L = 0L=3,ML​=0)\"\u003e\u003c/a\u003e5. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ef\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-Orbital (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL = 3, M_L = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3283em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/h3\u003e\u003cp\u003eComplex multi-lobed structure:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(7)/2*(5*cos(\u003cspan class=\"k\"\u003e\\tdplottheta\u003c/span\u003e)\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3 - 3*cos(\u003cspan class=\"k\"\u003e\\tdplottheta\u003c/span\u003e))\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry these modifications to customize your plot:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eAdjust Surface Opacity\u003c/strong\u003e:\nChange \u003ccode\u003efill opacity=.7\u003c/code\u003e to \u003ccode\u003e.9\u003c/code\u003e for a solid surface or \u003ccode\u003e.4\u003c/code\u003e to reveal internal grid lines.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eModify Camera Perspective\u003c/strong\u003e:\nChange \u003ccode\u003e\\tdplotsetmaincoords{60}{45}\u003c/code\u003e to view the surface from top or side angles.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe \u003ccode\u003etikz-3dplot\u003c/code\u003e package offers an elegant solution for rendering 3D spherical polar surface plots directly inside LaTeX without depending on external software like MATLAB or Python. Its parametric color fill capabilities make intuitive visualization of complex functions and quantum wave functions effortless.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eJeff Hein, \u003cem\u003eSpherical polar plots with 3dplot\u003c/em\u003e, \u003ca href=\"https://texample.net/spherical-polar-pots-with-3dplot/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eJeff Hein, \u003cem\u003eDocumentation for the 3dplot.sty package\u003c/em\u003e, \u003ca href=\"http://www.heinjd.com/dev/latex/3dplot/3dplot_documentation.pdf\" target=\"_blank\" rel=\"noopener noreferrer\"\u003e3dplot Manual\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eWikipedia, \u003cem\u003eSpherical Harmonics\u003c/em\u003e, \u003ca href=\"https://en.wikipedia.org/wiki/Spherical_harmonics\" target=\"_blank\" rel=\"noopener noreferrer\"\u003een.wikipedia.org/wiki/Spherical_harmonics\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Calculating C(n,4) Intersection Points of Chords in a Circle with TikZ", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cn4-points-of-intersection/", "url" : "https://www.aantriono.com/en/cn4-points-of-intersection/", "summary": "This article explores a problem in combinatorial geometry regarding intersection points of chords inside a circle and demonstrates how to implement its algorithm using TikZ in LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores a problem in combinatorial geometry regarding intersection points of chords inside a circle and demonstrates how to implement its algorithm using TikZ in LaTeX.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn \u003cstrong\u003ecombinatorial geometry\u003c/strong\u003e, there is a classic theorem regarding the relationship between points placed on a convex curve and the intersecting chords connecting them.\u003c/p\u003e\n\u003cp\u003eIf we place \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e points along the boundary of a circle (or any convex closed curve) and connect every pair of points with straight lines, we construct the \u003cstrong\u003ecomplete graph\u003c/strong\u003e \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. The question arises: \u003cstrong\u003eHow many intersection points are generated inside the circle by these line segments?\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eMathematically, the answer is precisely \u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{n}{4} = C(n,4)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7454em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/strong\u003e intersection points (assuming no three chords intersect at a single point).\u003c/p\u003e\n\u003cp\u003eThis article analyzes a TikZ implementation by \u003cstrong\u003eHugues Vermeiren\u003c/strong\u003e that utilizes a 4-combination generation algorithm to automatically locate and draw all internal intersection points.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"mathematical-foundation-why-\" class=\"headerLink\"\u003e\n \u003ca href=\"#mathematical-foundation-why-\" class=\"header-mark\" aria-label=\"Direct link to section: Mathematical Foundation: Why C(n,4)C(n,4)C(n,4)?\"\u003e\u003c/a\u003eMathematical Foundation: Why \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC(n,4)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e?\u003c/h2\u003e\u003cp\u003eTo understand why the number of intersection points is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC(n,4)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, consider the following geometric logic:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eAny \u003cstrong\u003efour distinct points\u003c/strong\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA, B, C, D\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) on the circle form a convex quadrilateral.\u003c/li\u003e\n\u003cli\u003eA convex quadrilateral has two diagonals, namely segments \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eThese two diagonals \u003cstrong\u003ealways intersect at exactly one point\u003c/strong\u003e inside the circle.\u003c/li\u003e\n\u003cli\u003eConversely, every single intersection point inside the circle is uniquely produced by the crossing of two diagonals from a unique set of 4 points.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eTherefore, the total number of intersection points inside the circle equals the number of ways to choose 4 points out of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, given by:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC(n,4) = \\binom{n}{4} = \\frac{n(n-1)(n-2)(n-3)}{24}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.113em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.427em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFor instance, with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en = 8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e points on a circle, the number of generated intersection points is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e70\u003c/mn\u003e\u003cmtext\u003e intersection points\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC(8,4) = \\frac{8 \\times 7 \\times 6 \\times 5}{24} = 70 \\text{ intersection points}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8623em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e70\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e intersection points\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete code by Hugues Vermeiren to draw the complete graph \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and mark all 70 intersection points automatically:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% C(n,4) points of intersection\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Hugues Vermeiren\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[active,tightpage]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\\PreviewBorder\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eifthen\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earrows,calc,intersections\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\r\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e8\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\myangles\u003c/span\u003e\u003cspan class=\"nb\"\u003e{{\u003c/span\u003e25,50,85,125,160,220,250,280,340\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e \u003cspan class=\"c\"\u003e% Position angles of points\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Variables and counters used to generate the 4-combinations\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003enp\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003enp\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e+1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ena\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003enb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003enc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eia\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ena\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e-1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003enb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e-2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003enc\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e-3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eq\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eq\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% exit flag\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ee\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ee\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% combination / intersection counter\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ea\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ea\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ec\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ec\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ed\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ed\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eConsider \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003en\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e randomly placed points on a circle.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e The complete graph on the \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003en\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e points has \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{pmatrix}n\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{pmatrix}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e edges.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Each pair of edges yields an intersection point and there are (at most) \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{pmatrix}n\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"m\"\u003e4\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{pmatrix}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e such points.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% 1. Draw the Complete Graph K_n\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=blue!10!green!10!,draw=blue,dotted,thick]\u003c/span\u003e (0,0) circle (\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e-1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\let\\nn\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,\u003cspan class=\"k\"\u003e\\nn\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e+1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\let\\ii\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\i]\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\let\\t\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\j\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ii\u003c/span\u003e,...,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\j]\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\let\\u\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue,very thick]\u003c/span\u003e (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e)--(\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\u\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\u\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% 2. Draw Node Points on the Circle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\i]\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\let\\t\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eia\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e+1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=blue,fill=blue!20!,thick]\u003c/span\u003e (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e) circle (2.5mm) node\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\mathbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\theia\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% 3. Loop Algorithm for C(n,4) Combinations \u0026amp; Intersection Plotting\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\whiledo\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\theq\u003c/span\u003e=0\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\stepcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ee\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thee\u003c/span\u003e=1000\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eq\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}}{}\u003c/span\u003e\u003cspan class=\"c\"\u003e% safety exit\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thed\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thec\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\thena\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\theb\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\thenb\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thea\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\thenc\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eq\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\stepcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ea\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thea\u003c/span\u003e+1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ec\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thea\u003c/span\u003e+2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ed\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thea\u003c/span\u003e+3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\stepcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ec\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\theb\u003c/span\u003e+1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ed\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\theb\u003c/span\u003e+2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\stepcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ec\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ed\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thec\u003c/span\u003e+1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\stepcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ed\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\theq\u003c/span\u003e=0\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Compute Coordinates for 4 Points\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\thea]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\xa\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\thea]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\ya\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\theb]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\xb\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\theb]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\yb\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\thec]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\xc\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\thec]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\yc\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\thed]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\xd\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\r*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\myangles\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\thed]\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\let\\yd\\pgfmathresult\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (A) at (\u003cspan class=\"k\"\u003e\\xa\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\ya\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (B) at (\u003cspan class=\"k\"\u003e\\xb\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\yb\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (C) at (\u003cspan class=\"k\"\u003e\\xc\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\yc\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (D) at (\u003cspan class=\"k\"\u003e\\xd\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\yd\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Define Diagonal Paths\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=sega]\u003c/span\u003e (A) -- (C);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=segb]\u003c/span\u003e (B) -- (D);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Compute Intersections via TikZ Intersections Library\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e [name intersections=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eof=sega and segb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e];\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (X) at (intersection-1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=green!50!,draw=blue]\u003c/span\u003e (X) circle (0.8mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}{}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\addtocounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ee\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e-1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNumber of generated intersection points : \u003cspan class=\"k\"\u003e\\thee\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"compiled-output\" class=\"headerLink\"\u003e\n \u003ca href=\"#compiled-output\" class=\"header-mark\" aria-label=\"Direct link to section: Compiled Output\"\u003e\u003c/a\u003eCompiled Output\u003c/h2\u003e\u003cp\u003eThe resulting graphical output displays a beautiful symmetrical web of lines punctuated by green dots at each intersection:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/cn4-points-of-intersection/intersection.webp\" title=\"C(n,4) Points of Intersection\" data-thumbnail=\"/cn4-points-of-intersection/intersection.webp\" data-sub-html=\"\u003ch2\u003eC(n,4) Intersection Points Output\u003c/h2\u003e\u003cp\u003eC(n,4) Points of Intersection\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/cn4-points-of-intersection/intersection.webp' srcset=\"/cn4-points-of-intersection/intersection_hu_f9f53e3ec0c2ea8e.webp 800w, /cn4-points-of-intersection/intersection_hu_39cd4d6c9cd105c.webp 1200w, /cn4-points-of-intersection/intersection_hu_387be13f9fdc350f.webp 1600w\" alt=\"C(n,4) Points of Intersection\" height=\"500\" width=\"476\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eC(n,4) Intersection Points Output\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Analysis\"\u003e\u003c/a\u003eDetailed Code Analysis\u003c/h2\u003e\u003ch3 id=\"1-key-libraries--package-setup\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-key-libraries--package-setup\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Key Libraries \u0026amp; Package Setup\"\u003e\u003c/a\u003e1. Key Libraries \u0026amp; Package Setup\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003eintersections\u003c/code\u003e\u003c/strong\u003e: An essential TikZ library (\u003ccode\u003e\\usetikzlibrary{intersections}\u003c/code\u003e) that automatically computes line-line intersections without requiring manual slope or intercept calculations.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003eifthen\u003c/code\u003e\u003c/strong\u003e: Provides control structures like \u003ccode\u003e\\ifthenelse\u003c/code\u003e and \u003ccode\u003e\\whiledo\u003c/code\u003e in standard LaTeX environments.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-building-the-complete-graph\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-building-the-complete-graph\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Building the Complete Graph KnK_nKn​\"\u003e\u003c/a\u003e2. Building the Complete Graph \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eThe array \u003ccode\u003e\\myangles\u003c/code\u003e stores the polar position of each of the 8 points.\u003c/li\u003e\n\u003cli\u003eNested loops \u003ccode\u003e\\foreach \\i\u003c/code\u003e and \u003ccode\u003e\\foreach \\j\u003c/code\u003e connect every pair \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ei\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6595em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ej\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ej\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.854em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05724em;\"\u003ej\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmi\u003ej\u003c/mi\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0 \\le i \u0026lt; j \\le n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7804em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6986em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.854em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05724em;\"\u003ej\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) with blue lines \u003ccode\u003e\\draw[blue,very thick]\u003c/code\u003e. The total number of drawn chords is:\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e28\u003c/mn\u003e\u003cmtext\u003e edges\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{8}{2} = \\frac{8 \\times 7}{2} = 28 \\text{ edges}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2451em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8951em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e×\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e28\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e edges\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-the-4-combination-generator\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-the-4-combination-generator\" class=\"header-mark\" aria-label=\"Direct link to section: 3. The 4-Combination Generator\"\u003e\u003c/a\u003e3. The 4-Combination Generator\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eThe loop \u003ccode\u003e\\whiledo{\\theq=0}\u003c/code\u003e implements a lexicographical combination algorithm to produce tuples \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a, b, c, d)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e such that \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \u0026lt; b \u0026lt; c \u0026lt; d\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7335em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eThis guarantees that every set of 4 points is evaluated exactly once.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"4-interactive-intersection-finding\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-interactive-intersection-finding\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Interactive Intersection Finding\"\u003e\u003c/a\u003e4. Interactive Intersection Finding\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eFor each 4-point set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(A, B, C, D)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, two diagonal paths \u003ccode\u003esega\u003c/code\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e→\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA \\to C\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e→\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) and \u003ccode\u003esegb\u003c/code\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e→\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB \\to D\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e→\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) are generated without drawing.\u003c/li\u003e\n\u003cli\u003eCommand \u003ccode\u003e\\path [name intersections={of=sega and segb}];\u003c/code\u003e computes the meeting point and names it \u003ccode\u003e(intersection-1)\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eA green dot is plotted at coordinate \u003ccode\u003e(X)\u003c/code\u003e via \u003ccode\u003e\\fill[fill=green!50!,draw=blue] (X) circle (0.8mm);\u003c/code\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry these modifications to explore further:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eVarying Point Count (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/strong\u003e:\nChange \u003ccode\u003e\\def\\n{8}\u003c/code\u003e to \u003ccode\u003e\\def\\n{6}\u003c/code\u003e for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{6}{4} = 15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2451em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8951em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e points, or \u003ccode\u003e\\def\\n{10}\u003c/code\u003e for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e210\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{10}{4} = 210\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2451em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8951em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e210\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e points.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eRegular Polygon Placement\u003c/strong\u003e:\nReplace the manual \u003ccode\u003e\\myangles\u003c/code\u003e vector with uniform angular steps:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e * 360 / \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eRadial Color Mapping\u003c/strong\u003e:\nColor intersection points according to their radial distance from the origin for a vibrant aesthetic effect.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThis TikZ script beautifully bridges \u003cstrong\u003ecombinatorial mathematics\u003c/strong\u003e and \u003cstrong\u003ecomputational geometry\u003c/strong\u003e. By leveraging TikZ\u0026rsquo;s \u003ccode\u003eintersections\u003c/code\u003e library, the exact physical positions of dozens of chord intersections are generated automatically from elementary combination rules.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eHugues Vermeiren, \u003cem\u003ePoints of intersection\u003c/em\u003e, \u003ca href=\"https://texample.net/cn4-points-of-intersection/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003cli\u003eGraham, R. L., Knuth, D. E., \u0026amp; Patashnik, O., \u003cem\u003eConcrete Mathematics: A Foundation for Computer Science\u003c/em\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Creating a 2026 Circular Year Calendar using the TikZ Calendar Library in LaTeX", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/calendar-circles/", "url" : "https://www.aantriono.com/en/calendar-circles/", "summary": "This article explores design techniques for creating a 2026 circular infographic calendar using the calendar and shadings libraries in LaTeX TikZ.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores design techniques for creating a 2026 circular infographic calendar using the \u003ccode\u003ecalendar\u003c/code\u003e and \u003ccode\u003eshadings\u003c/code\u003e libraries in LaTeX TikZ.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eTraditional wall or desk calendars typically rely on rigid, square 12-month table grids. However, by taking advantage of LaTeX\u0026rsquo;s \u003cstrong\u003eTikZ\u003c/strong\u003e package and its built-in \u003ccode\u003ecalendar\u003c/code\u003e library, we can craft an artistic \u003cstrong\u003ecircular infographic calendar\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eOriginally conceived by \u003cstrong\u003eTill Tantau\u003c/strong\u003e (author of PGF/TikZ) and refined by \u003cstrong\u003eStefan Kottwitz\u003c/strong\u003e, this layout features a prominent central year node surrounded by 12 radial month circles.\u003c/p\u003e\n\u003cp\u003eIn this guide, all dates and day-of-the-week calculations are tailored specifically for the \u003cstrong\u003eYear 2026\u003c/strong\u003e (where January 1, 2026 falls on a Thursday, February 2026 has 28 days, etc.).\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code-year-2026\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code-year-2026\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code (Year 2026)\"\u003e\u003c/a\u003eLaTeX Source Code (Year 2026)\u003c/h2\u003e\u003cp\u003eHere is the complete code customized for the \u003cstrong\u003eYear 2026\u003c/strong\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% A calendar of circles - Year 2026\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Till Tantau (The PGF manual)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Modifications: Stefan Kottwitz \u0026amp; Cleanups\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[active,tightpage]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\PreviewEnvironment\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\\PreviewBorder\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalendar,shadings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Default Sans-Serif Font\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\renewcommand*\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\familydefault\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\sfdefault\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% 4-Season Color Scheme\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewinter\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblue!60!cyan\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003espring\u003cspan class=\"nb\"\u003e}{\u003c/span\u003egreen!60!black\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esummer\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eorange!85!black\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efall\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ered!80!black\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% TeX Counter for Radial Angle Computation\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcount\\mycount\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Set Target Year to 2026\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\year\u003c/span\u003e=2026\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[every day/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eanchor=mid,font=\u003cspan class=\"k\"\u003e\\tiny\\bfseries\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 1. Lingkaran Besar Paling Luar - DIUBAH MENJADI BIRU SOLID\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!35]\u003c/span\u003e (0,0) circle (7.4cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 2. Lingkaran Gradasi di Tengah (Biru di pinggir, putih di pusat)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[shading=radial, outer color=blue!60!cyan!80, inner color=white, draw=none]\u003c/span\u003e (0,0) circle (2.9cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 3. Teks Tahun \u0026#34;2026\u0026#34; tepat di Pusat\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[circle, draw=none, font=\\fontfamily{phv}\\selectfont\\Huge\\bfseries]\u003c/span\u003e at (0,0) \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue!90!black\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\the\\year\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 4. Loop Melalui 12 Bulan dengan Gradasi Lembut (Putih di Tengah Bulan)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\month\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\monthcolor\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\monthname\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 1/winter/January, 2/winter/February, 3/spring/March, 4/spring/April, 5/spring/May, 6/summer/June, \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 7/summer/July, 8/summer/August, 9/fall/September, 10/fall/October, 11/fall/November, 12/gray!70!black/December\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Hitung posisi sudut radial (Januari di posisi jam 6 / -90 derajat)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\mycount\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\month\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by -1\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\multiply\\mycount\u003c/span\u003e by 30\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by -90\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Koordinat pusat lingkaran bulan\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (M) at (\u003cspan class=\"k\"\u003e\\the\\mycount\u003c/span\u003e:5.4cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Gambar Lingkaran Bulan (Gradasi memudar dari luar ke dalam putih)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\shadedraw\u003c/span\u003e[shading=radial, \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e outer color=\u003cspan class=\"k\"\u003e\\monthcolor\u003c/span\u003e!45!white, \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e inner color=white, \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e draw=none] \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (M) circle(1.4cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Cetak Nama Bulan Tepat di Pusat Lingkaran Kecil yang Putih\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[font=\\large\\sffamily\\bfseries, color=\\monthcolor!90!black]\u003c/span\u003e at (M) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\monthname\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Plot Angka Kalender 2026 mengelilingi nama bulan\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e at (M) [\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e dates=\u003cspan class=\"k\"\u003e\\the\\year\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\month\u003c/span\u003e-01 to \u003cspan class=\"k\"\u003e\\the\\year\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\month\u003c/span\u003e-last\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e if (Sunday) [red] \u003cspan class=\"c\"\u003e% PENTING: pakai kurung SIKU, bukan kurung kurawal,\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% supaya \u0026#34;red\u0026#34; dibaca sebagai opsi warna TikZ,\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% bukan sebagai kode TeX biasa.\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e if (all) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Mengatur rotasi penempatan tanggal agar melingkar pas\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\mycount\u003c/span\u003e=1\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by -\u003cspan class=\"k\"\u003e\\pgfcalendarcurrentday\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\multiply\\mycount\u003c/span\u003e by 11\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by 90\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgftransformshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfpointpolar\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\mycount\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1.2cm\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"output-produced-2026-calendar\" class=\"headerLink\"\u003e\n \u003ca href=\"#output-produced-2026-calendar\" class=\"header-mark\" aria-label=\"Direct link to section: Output Produced (2026 Calendar)\"\u003e\u003c/a\u003eOutput Produced (2026 Calendar)\u003c/h2\u003e\u003cp\u003eThe resulting graphical output renders a stunning circular calendar for 2026:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/calendar-circles/calendar-circles.webp\" title=\"2026 Circular Calendar TikZ\" data-thumbnail=\"/calendar-circles/calendar-circles.webp\" data-sub-html=\"\u003ch2\u003e2026 Circular Calendar Output\u003c/h2\u003e\u003cp\u003e2026 Circular Calendar TikZ\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/calendar-circles/calendar-circles.webp' srcset=\"/calendar-circles/calendar-circles_hu_32365d83c8403e91.webp 800w, /calendar-circles/calendar-circles_hu_79e0b03af6e30242.webp 1200w, /calendar-circles/calendar-circles_hu_a3c15f8d197d56bd.webp 1600w\" alt=\"2026 Circular Calendar TikZ\" height=\"712\" width=\"711\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003e2026 Circular Calendar Output\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-analysis--logic\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-analysis--logic\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Analysis \u0026amp; Logic\"\u003e\u003c/a\u003eDetailed Code Analysis \u0026amp; Logic\u003c/h2\u003e\u003ch3 id=\"1-the-calendar-library--2026-date-calculations\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-the-calendar-library--2026-date-calculations\" class=\"header-mark\" aria-label=\"Direct link to section: 1. The calendar Library \u0026amp; 2026 Date Calculations\"\u003e\u003c/a\u003e1. The \u003ccode\u003ecalendar\u003c/code\u003e Library \u0026amp; 2026 Date Calculations\u003c/h3\u003e\u003cp\u003eTikZ\u0026rsquo;s \u003ccode\u003ecalendar\u003c/code\u003e library automatically computes weekday alignment for any given month in \u003ccode\u003e2026\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003edates=2026-\u003cspan class=\"k\"\u003e\\month\u003c/span\u003e-01 to 2026-\u003cspan class=\"k\"\u003e\\month\u003c/span\u003e-last\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe engine automatically handles:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eJanuary 2026 starting on a Thursday with 31 days.\u003c/li\u003e\n\u003cli\u003eFebruary 2026 having 28 days (non-leap year).\u003c/li\u003e\n\u003cli\u003eHighlighting all Sundays in red via \u003ccode\u003eif (Sunday) [red]\u003c/code\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-4-season-color-palette\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-4-season-color-palette\" class=\"header-mark\" aria-label=\"Direct link to section: 2. 4-Season Color Palette\"\u003e\u003c/a\u003e2. 4-Season Color Palette\u003c/h3\u003e\u003cp\u003eMonth background shading is grouped by season:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eWinter\u003c/strong\u003e: Blue (\u003ccode\u003eblue\u003c/code\u003e) for months 1 (Jan), 2 (Feb), and 12 (Dec).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSpring\u003c/strong\u003e: Deep Green (\u003ccode\u003egreen!60!black\u003c/code\u003e) for months 3 (Mar), 4 (Apr), and 5 (May).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSummer\u003c/strong\u003e: Warm Orange (\u003ccode\u003eorange\u003c/code\u003e) for months 6 (Jun), 7 (Jul), and 8 (Aug).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFall\u003c/strong\u003e: Rich Red (\u003ccode\u003ered\u003c/code\u003e) for months 9 (Sep), 10 (Oct), and 11 (Nov).\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-radial-distribution-of-12-months\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-radial-distribution-of-12-months\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Radial Distribution of 12 Months\"\u003e\u003c/a\u003e3. Radial Distribution of 12 Months\u003c/h3\u003e\u003cp\u003eEach month node is positioned at a radius of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5.4\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5.4\\text{ cm}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5.4\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e from the origin at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e30^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e increments (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmi mathvariant=\"normal\"\u003e/\u003c/mi\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^\\circ / 12 = 30^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e/12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e):\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eMonth Angle\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmtext\u003emonth\u003c/mtext\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\text{Month Angle} = (\\text{month} - 1) \\times 30^\\circ - 90^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eMonth Angle\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003emonth\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8074em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"4-circular-ring-alignment-of-day-numbers\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-circular-ring-alignment-of-day-numbers\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Circular Ring Alignment of Day Numbers\"\u003e\u003c/a\u003e4. Circular Ring Alignment of Day Numbers\u003c/h3\u003e\u003cp\u003eInside each month node, day numbers (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e…\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e31\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1, 2, \\dots, 31\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e31\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) are arranged in a ring at radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1.2\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1.2\\text{ cm}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1.2\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e using polar transforms:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eDay Angle\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmtext\u003ecurrent_day\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\text{Day Angle} = (1 - \\text{current\\_day}) \\times 11^\\circ + 90^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eDay Angle\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.06em;vertical-align:-0.31em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecurrent_day\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8074em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry these modifications to customize your calendar:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eHighlighting Holidays / Special Dates\u003c/strong\u003e:\nAdd conditional styling inside the \u003ccode\u003e\\calendar\u003c/code\u003e options, for example:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eif (equals=2026-12-25) [red, font=\u003cspan class=\"k\"\u003e\\bfseries\u003c/span\u003e] \u003cspan class=\"err\"\u003e%\u003c/span\u003e Christmas Day\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCustom Color Gradients\u003c/strong\u003e:\nExperiment with different color themes such as pastel or dark mode palettes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe \u003ccode\u003ecalendar\u003c/code\u003e library in TikZ LaTeX offers a powerful engine for building automated, custom-tailored calendars for \u003cstrong\u003e2026\u003c/strong\u003e. Combining automated date calculations with radial shading yields a premium infographic graphic.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eTill Tantau \u0026amp; Stefan Kottwitz, \u003cem\u003eA calendar of circles\u003c/em\u003e, \u003ca href=\"https://texample.net/calendar-circles/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual (Calendar Library)\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Drawing a Directed Complete Graph (K16) with TikZ in LaTeX", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/complete-graph/", "url" : "https://www.aantriono.com/en/complete-graph/", "summary": "This article explores Graph Theory concepts and LaTeX TikZ coding techniques to render a Directed Complete Graph with 16 nodes (K16K_{16}K16​).\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores Graph Theory concepts and LaTeX TikZ coding techniques to render a Directed Complete Graph with 16 nodes (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_{16}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn \u003cstrong\u003eGraph Theory\u003c/strong\u003e, a \u003cstrong\u003ecomplete graph\u003c/strong\u003e \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is defined as a simple graph where every pair of distinct vertices is connected by a unique edge.\u003c/p\u003e\n\u003cp\u003eWhen extended to a \u003cstrong\u003edirected complete graph\u003c/strong\u003e (or symmetric complete digraph), every pair of vertices \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eu\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eu\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ev\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ev\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is connected by \u003cstrong\u003etwo opposing directed edges\u003c/strong\u003e: one arrow pointing from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eu\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eu\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ev\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ev\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and another pointing from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ev\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ev\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eu\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eu\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThis article breaks down a LaTeX TikZ implementation by \u003cstrong\u003eQuintin Jean-Noël\u003c/strong\u003e. The code leverages nested loops, TeX counter arithmetic for circular polar positioning, and subtle curved paths (\u003ccode\u003ebend left\u003c/code\u003e and \u003ccode\u003ebend right\u003c/code\u003e) to cleanly render 240 directed edges connecting 16 nodes.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"mathematical-analysis-complete-graph\" class=\"headerLink\"\u003e\n \u003ca href=\"#mathematical-analysis-complete-graph\" class=\"header-mark\" aria-label=\"Direct link to section: Mathematical Analysis: Complete Graph K16K_{16}K16​\"\u003e\u003c/a\u003eMathematical Analysis: Complete Graph \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_{16}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h2\u003e\u003cp\u003eFor a directed complete graph with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en = 16\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e vertices:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eNumber of Vertices (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eV\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/strong\u003e:\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e|V| = 16\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNumber of Unique Vertex Pairs\u003c/strong\u003e:\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e120\u003c/mn\u003e\u003cmtext\u003e pairs\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{16}{2} = \\frac{16 \\times 15}{2} = 120 \\text{ pairs}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2451em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8951em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e16\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e×\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8623em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e120\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e pairs\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTotal Directed Edges (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/strong\u003e:\nSince each pair has 2 anti-parallel directed edges:\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e∣\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e240\u003c/mn\u003e\u003cmtext\u003e directed edges\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e|E| = 2 \\times \\binom{16}{2} = 16 \\times 15 = 240 \\text{ directed edges}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord\"\u003e∣\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2451em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8951em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e240\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e directed edges\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eVertex Degree\u003c/strong\u003e:\nEach vertex is connected to the remaining 15 vertices:\n\u003cul\u003e\n\u003cli\u003eOut-degree: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed^+(v) = 15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0213em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7713em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ev\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eIn-degree: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed^-(v) = 15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0213em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7713em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ev\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete TikZ code by Quintin Jean-Noël:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% A complete graph\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Quintin Jean-Noël\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[active,tightpage]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\PreviewEnvironment\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\\PreviewBorder\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"na\"\u003e[topaths]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% TeX counter for precise angle calculation\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcount\\mycount\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[transform shape]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 1. Create First 8 Nodes (Angles: 0, 45, 90, 135, ...)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\number\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,8\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\mycount\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\number\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by -1\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\multiply\\mycount\u003c/span\u003e by 45\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by 0\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw,circle,inner sep=0.25cm]\u003c/span\u003e (N-\u003cspan class=\"k\"\u003e\\number\u003c/span\u003e) at (\u003cspan class=\"k\"\u003e\\the\\mycount\u003c/span\u003e:5.4cm) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 2. Create Second 8 Nodes with 22.5 Degree Offset (Angles: 22.5, 67.5, ...)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\number\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e9,...,16\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\mycount\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\number\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by -1\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\multiply\\mycount\u003c/span\u003e by 45\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by 22.5\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw,circle,inner sep=0.25cm]\u003c/span\u003e (N-\u003cspan class=\"k\"\u003e\\number\u003c/span\u003e) at (\u003cspan class=\"k\"\u003e\\the\\mycount\u003c/span\u003e:5.4cm) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 3. Connect All Vertex Pairs with Opposing Curved Edges\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\number\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,15\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\mycount\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\number\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\advance\\mycount\u003c/span\u003e by 1\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\numbera\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\the\\mycount\u003c/span\u003e,...,16\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e (N-\u003cspan class=\"k\"\u003e\\number\u003c/span\u003e) edge[-\u0026gt;,bend right=3] (N-\u003cspan class=\"k\"\u003e\\numbera\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e edge[\u0026lt;-,bend left=3] (N-\u003cspan class=\"k\"\u003e\\numbera\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"output-produced\" class=\"headerLink\"\u003e\n \u003ca href=\"#output-produced\" class=\"header-mark\" aria-label=\"Direct link to section: Output Produced\"\u003e\u003c/a\u003eOutput Produced\u003c/h2\u003e\u003cp\u003eThe resulting graphical output displays a dense, symmetrical circular graph network:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/complete-graph/complete-graph.webp\" title=\"Complete Graph K16\" data-thumbnail=\"/complete-graph/complete-graph.webp\" data-sub-html=\"\u003ch2\u003eComplete Graph K16 TikZ Output\u003c/h2\u003e\u003cp\u003eComplete Graph K16\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/complete-graph/complete-graph.webp' srcset=\"/complete-graph/complete-graph_hu_7c88f0ba1fec0eb5.webp 800w, /complete-graph/complete-graph_hu_139460d9637dd17d.webp 1200w, /complete-graph/complete-graph_hu_2ba06a1c8986539e.webp 1600w\" alt=\"Complete Graph K16\" height=\"500\" width=\"500\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eComplete Graph K16 TikZ Output\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-analysis--logic\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-analysis--logic\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Analysis \u0026amp; Logic\"\u003e\u003c/a\u003eDetailed Code Analysis \u0026amp; Logic\u003c/h2\u003e\u003ch3 id=\"1-tex-counter-arithmetic-for-angular-placement\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-tex-counter-arithmetic-for-angular-placement\" class=\"header-mark\" aria-label=\"Direct link to section: 1. TeX Counter Arithmetic for Angular Placement\"\u003e\u003c/a\u003e1. TeX Counter Arithmetic for Angular Placement\u003c/h3\u003e\u003cp\u003eTraditional TeX loops do not support floating-point arithmetic directly inside native loop expressions. To overcome this, the author splits the 16 nodes into two passes using \u003ccode\u003e\\newcount\\mycount\u003c/code\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePass 1 (Nodes 1–8)\u003c/strong\u003e: Computes angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmtext\u003enumber\u003c/mtext\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e45\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(\\text{number} - 1) \\times 45^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003enumber\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, producing \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e45\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e135\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e225\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e270\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e315\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0^\\circ, 45^\\circ, 90^\\circ, 135^\\circ, 180^\\circ, 225^\\circ, 270^\\circ, 315^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8686em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e27\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e31\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePass 2 (Nodes 9–16)\u003c/strong\u003e: Adds a \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e22.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e22.5^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e22.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e fractional offset, producing \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e22.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e67.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e112.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e157.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e202.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e247.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e292.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e337.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e22.5^\\circ, 67.5^\\circ, 112.5^\\circ, 157.5^\\circ, 202.5^\\circ, 247.5^\\circ, 292.5^\\circ, 337.5^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8686em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e22.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e67.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e112.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e157.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e202.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e247.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e292.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e337.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eEach node is assigned a TikZ identifier \u003ccode\u003e(N-1)\u003c/code\u003e through \u003ccode\u003e(N-16)\u003c/code\u003e at a radial distance of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5.4\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5.4\\text{ cm}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5.4\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"2-nested-loop-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-nested-loop-structure\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Nested Loop Structure\"\u003e\u003c/a\u003e2. Nested Loop Structure\u003c/h3\u003e\u003cp\u003eTo connect every distinct pair of nodes without duplication:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThe outer loop runs from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ei = 1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6595em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eThe inner loop runs from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ej\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ej = i+1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.854em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05724em;\"\u003ej\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7429em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e16\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eThis executes exactly \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e120\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{16}{2} = 120\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2451em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8951em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e120\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e iterations.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-edge-separation-via-bend-left--bend-right\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-edge-separation-via-bend-left--bend-right\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Edge Separation via bend left \u0026amp; bend right\"\u003e\u003c/a\u003e3. Edge Separation via \u003ccode\u003ebend left\u003c/code\u003e \u0026amp; \u003ccode\u003ebend right\u003c/code\u003e\u003c/h3\u003e\u003cp\u003eIf two opposing directed arrows are drawn as straight lines over each other, they overlap completely and hide directional detail.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eOption \u003ccode\u003eedge[-\u0026gt;,bend right=3]\u003c/code\u003e curves the forward arrow \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to the right.\u003c/li\u003e\n\u003cli\u003eOption \u003ccode\u003eedge[\u0026lt;-,bend left=3]\u003c/code\u003e curves the return arrow \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to the left.\u003c/li\u003e\n\u003cli\u003eThis subtle curvature creates a small gap between anti-parallel edges, ensuring all 240 directed arrows remain distinctly visible.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry these modifications to explore further:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eVarying Vertex Count (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/strong\u003e:\nTry \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en = 6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en = 10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_{10}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) for a sparser network graph.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDistance-Based Edge Coloration\u003c/strong\u003e:\nColor edges based on chord length (e.g. blue for adjacent neighbors, red for diameter chords).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBipartite Graph Layout\u003c/strong\u003e:\nArrange nodes in two parallel columns to render a Complete Bipartite Graph (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK_{m,n}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003cspan class=\"mpunct mtight\"\u003e,\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eQuintin Jean-Noël\u0026rsquo;s TikZ code provides an excellent example of how abstract Graph Theory objects can be translated into clean visual representations in LaTeX. By leveraging counter calculations and curvature options (\u003ccode\u003ebend\u003c/code\u003e), 240 directed edges are rendered automatically with precision and balance.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eQuintin Jean-Noël, \u003cem\u003eA complete graph\u003c/em\u003e, \u003ca href=\"https://texample.net/complete-graph/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003cli\u003eDouglas B. West, \u003cem\u003eIntroduction to Graph Theory\u003c/em\u003e, 2nd Edition, Prentice Hall.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Drawing a Sunflower Pattern (Phyllotaxis) with TikZ in LaTeX", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/phyllotaxy/", "url" : "https://www.aantriono.com/en/phyllotaxy/", "summary": "This article presents an aesthetic analysis and a detailed mathematical breakdown of how to draw a sunflower (phyllotaxis) pattern using the TikZ package in LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article presents an aesthetic analysis and a detailed mathematical breakdown of how to draw a sunflower (phyllotaxis) pattern using the TikZ package in LaTeX.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cstrong\u003ePhyllotaxis\u003c/strong\u003e (or phyllotaxy) is the arrangement of leaves, petals, or seeds on a plant. One of the most fascinating phenomena in nature is the spiral pattern found on a sunflower head. This pattern is not random; it follows precise mathematical rules involving the \u003cstrong\u003eGolden Ratio\u003c/strong\u003e and the \u003cstrong\u003eGolden Angle\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eIn LaTeX-based graphics, we can reproduce this natural beauty with high precision using the \u003cstrong\u003eTikZ\u003c/strong\u003e package. This article explores a TikZ implementation by \u003cstrong\u003eDavid Convent\u003c/strong\u003e, which adapts Jim Bumgardner\u0026rsquo;s tutorial \u003cem\u003e\u0026ldquo;Circles, Spirals and Sunflowers\u0026rdquo;\u003c/em\u003e. We will dissect how mathematical logic is translated into LaTeX code to generate a stunning phyllotaxis visualization.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code to generate the phyllotaxis pattern using TikZ:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Phyllotaxy\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: David Convent\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz,border=10pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\nbrcircles\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e377\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\outerradius\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e30mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\deviation\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.9\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\fudge\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.62\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[scale=.32]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\goldenRatio\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e(1+sqrt(5))\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\meanArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003epow(\u003cspan class=\"k\"\u003e\\outerradius\u003c/span\u003e * 10 / \u003cspan class=\"k\"\u003e\\nbrcircles\u003c/span\u003e, 2) * pi\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\minArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\meanArea\u003c/span\u003e * (1 - \u003cspan class=\"k\"\u003e\\deviation\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\midArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\meanArea\u003c/span\u003e * (1 + \u003cspan class=\"k\"\u003e\\deviation\u003c/span\u003e) - \u003cspan class=\"k\"\u003e\\minArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\b\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,\u003cspan class=\"k\"\u003e\\nbrcircles\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% mod() must be used in order to calculate the right angle.\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% otherwise, when \\b is greater than 28 the angle is greater\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% than 16384 and an error is raised (\u0026#39;Dimension too large\u0026#39;).\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% -- thx Tonio for this one.\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\angle\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003emod(\u003cspan class=\"k\"\u003e\\goldenRatio\u003c/span\u003e * \u003cspan class=\"k\"\u003e\\b\u003c/span\u003e, 2) * 180\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\sratio\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\b\u003c/span\u003e / \u003cspan class=\"k\"\u003e\\nbrcircles\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\smArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\minArea\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\sratio\u003c/span\u003e * \u003cspan class=\"k\"\u003e\\midArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\smRadius\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt(\u003cspan class=\"k\"\u003e\\smArea\u003c/span\u003e / pi) / 2 * \u003cspan class=\"k\"\u003e\\fudge\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\addtocounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\smArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(\u003cspan class=\"k\"\u003e\\value\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}\u003c/span\u003e / pi) / 2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\angle\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\pgfmathresult\u003c/span\u003e) circle [radius=\u003cspan class=\"k\"\u003e\\smRadius\u003c/span\u003e] ;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"output-produced\" class=\"headerLink\"\u003e\n \u003ca href=\"#output-produced\" class=\"header-mark\" aria-label=\"Direct link to section: Output Produced\"\u003e\u003c/a\u003eOutput Produced\u003c/h2\u003e\u003cp\u003eWhen compiled, the code above yields the following spiral pattern:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/phyllotaxy/phyllotaxy.webp\" title=\"Sunflower Pattern (Phyllotaxis)\" data-thumbnail=\"/phyllotaxy/phyllotaxy.webp\" data-sub-html=\"\u003ch2\u003eSunflower Pattern (Phyllotaxis)\u003c/h2\u003e\u003cp\u003eSunflower Pattern (Phyllotaxis)\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/phyllotaxy/phyllotaxy.webp' srcset=\"/phyllotaxy/phyllotaxy_hu_513487d90311dadc.webp 800w, /phyllotaxy/phyllotaxy_hu_bba8fb1abe7c095e.webp 1200w, /phyllotaxy/phyllotaxy_hu_1ac457bde49750bf.webp 1600w\" alt=\"Sunflower Pattern (Phyllotaxis)\" height=\"493\" width=\"500\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSunflower Pattern (Phyllotaxis)\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-structure-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-structure-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Structure Explanation\"\u003e\u003c/a\u003eDetailed Code Structure Explanation\u003c/h2\u003e\u003ch3 id=\"1-initialization-and-libraries\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-initialization-and-libraries\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Initialization and Libraries\"\u003e\u003c/a\u003e1. Initialization and Libraries\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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Parameter Constants\"\u003e\u003c/a\u003e2. Parameter Constants\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\nbrcircles\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e377\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\outerradius\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e30mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\deviation\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.9\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\fudge\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.62\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\nbrcircles\u003c/code\u003e\u003c/strong\u003e: The total number of circles (seeds) to draw. The number \u003cstrong\u003e377\u003c/strong\u003e is a \u003cstrong\u003eFibonacci number\u003c/strong\u003e, which naturally appears in plant spiral patterns.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\outerradius\u003c/code\u003e\u003c/strong\u003e: The outer boundary radius of the overall design (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmtext\u003e mm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e30\\text{ mm}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e mm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\deviation\u003c/code\u003e\u003c/strong\u003e: A value (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0.9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0.9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) controlling how quickly the seed circles grow in size as they move outward.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\fudge\u003c/code\u003e\u003c/strong\u003e: A scaling factor (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0.62\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0.62\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.62\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) to pack or space out the circles nicely without excessive overlapping.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-cumulative-area-counter\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-cumulative-area-counter\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Cumulative Area Counter\"\u003e\u003c/a\u003e3. Cumulative Area Counter\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eDefines a LaTeX counter named \u003ccode\u003ecumulArea\u003c/code\u003e to track the sum of the areas of all circles drawn so far. This counter helps calculate the radial distance of each circle from the center.\u003c/p\u003e\n\u003ch3 id=\"4-core-geometric-calculations\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-core-geometric-calculations\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Core Geometric Calculations\"\u003e\u003c/a\u003e4. Core Geometric Calculations\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\goldenRatio\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e(1+sqrt(5))\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\meanArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003epow(\u003cspan class=\"k\"\u003e\\outerradius\u003c/span\u003e * 10 / \u003cspan class=\"k\"\u003e\\nbrcircles\u003c/span\u003e, 2) * pi\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\minArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\meanArea\u003c/span\u003e * (1 - \u003cspan class=\"k\"\u003e\\deviation\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\midArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\meanArea\u003c/span\u003e * (1 + \u003cspan class=\"k\"\u003e\\deviation\u003c/span\u003e) - \u003cspan class=\"k\"\u003e\\minArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\goldenRatio\u003c/code\u003e\u003c/strong\u003e: Defined as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1 + \\sqrt{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, which equals \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003eΦ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\Phi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2Φ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (twice the Golden Ratio).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\meanArea\u003c/code\u003e\u003c/strong\u003e: The average area of each small circle.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\minArea\u003c/code\u003e\u003c/strong\u003e and \u003cstrong\u003e\u003ccode\u003e\\midArea\u003c/code\u003e\u003c/strong\u003e: Lower bounds and scaling parameters to grow the circle size linearly from the center outwards.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"5-loop-iteration-and-angle-calculation\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-loop-iteration-and-angle-calculation\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Loop Iteration and Angle Calculation\"\u003e\u003c/a\u003e5. Loop Iteration and Angle Calculation\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\b\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,\u003cspan class=\"k\"\u003e\\nbrcircles\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\angle\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003emod(\u003cspan class=\"k\"\u003e\\goldenRatio\u003c/span\u003e * \u003cspan class=\"k\"\u003e\\b\u003c/span\u003e, 2) * 180\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis is where the mathematical magic happens. Each seed is placed at an angle that is a multiple of the Golden Angle.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThe expression \u003ccode\u003emod(\\goldenRatio * \\b, 2) * 180\u003c/code\u003e multiplies the seed index \u003ccode\u003e\\b\u003c/code\u003e by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003eΦ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\Phi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2Φ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, takes the modulo \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and scales the result (in the range \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e[0, 2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e[\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to get a degree angle in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e[0^\\circ, 360^\\circ)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e[\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eThe use of \u003ccode\u003emod()\u003c/code\u003e is crucial. Without it, the cumulative angle would exceed TeX\u0026rsquo;s internal coordinate limit of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e16384\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e16384\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16384\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e after just 28 seeds, throwing a \u003cem\u003e\u0026ldquo;Dimension too large\u0026rdquo;\u003c/em\u003e error.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"6-calculating-seed-size\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-calculating-seed-size\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Calculating Seed Size\"\u003e\u003c/a\u003e6. Calculating Seed Size\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\sratio\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\b\u003c/span\u003e / \u003cspan class=\"k\"\u003e\\nbrcircles\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\smArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\minArea\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\sratio\u003c/span\u003e * \u003cspan class=\"k\"\u003e\\midArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\smRadius\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt(\u003cspan class=\"k\"\u003e\\smArea\u003c/span\u003e / pi) / 2 * \u003cspan class=\"k\"\u003e\\fudge\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\sratio\u003c/code\u003e\u003c/strong\u003e: The ratio of the current seed\u0026rsquo;s index to the total number of seeds (ranging from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e at the center to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e at the boundary).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\smArea\u003c/code\u003e\u003c/strong\u003e: Calculates the area of the current seed linearly. Outer seeds have larger areas than inner ones.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\smRadius\u003c/code\u003e\u003c/strong\u003e: Converts the area back to a circle radius (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmtext\u003eArea\u003c/mtext\u003e\u003cmi mathvariant=\"normal\"\u003e/\u003c/mi\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er = \\sqrt{\\text{Area}/\\pi}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.305em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.935em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eArea\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e/\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.895em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.305em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) and scales it by \u003ccode\u003e\\fudge\u003c/code\u003e for aesthetic spacing.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"7-radial-position-accumulation-and-drawing\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-radial-position-accumulation-and-drawing\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Radial Position Accumulation and Drawing\"\u003e\u003c/a\u003e7. Radial Position Accumulation and Drawing\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\addtocounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\smArea\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathparse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(\u003cspan class=\"k\"\u003e\\value\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecumulArea\u003cspan class=\"nb\"\u003e}\u003c/span\u003e / pi) / 2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\angle\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\pgfmathresult\u003c/span\u003e) circle [radius=\u003cspan class=\"k\"\u003e\\smRadius\u003c/span\u003e] ;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\addtocounter{cumulArea}{\\smArea}\u003c/code\u003e: Adds the current seed\u0026rsquo;s area to the running total.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\pgfmathparse{...}\u003c/code\u003e: Calculates the radial distance from the center as the square root of the cumulative area divided by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. This ensures a \u003cstrong\u003eperfectly uniform distribution density\u003c/strong\u003e across the entire spiral.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\fill[]\u003c/code\u003e: Draws a black circle at polar coordinates \u003ccode\u003e(\\angle:\\pgfmathresult)\u003c/code\u003e with the radius \u003ccode\u003e\\smRadius\u003c/code\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"visual-and-mathematical-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-and-mathematical-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual and Mathematical Analysis\"\u003e\u003c/a\u003eVisual and Mathematical Analysis\u003c/h2\u003e\u003cp\u003eThe generated pattern highlights nature\u0026rsquo;s geometry:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eThe Golden Angle (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e137.5\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e137.5^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e137.\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e)\u003c/strong\u003e: This irrational angle is the most efficient way to partition circular space. It prevents the seeds from aligning in straight radial lines, avoiding gaps.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFibonacci Sequences\u003c/strong\u003e: The clockwise and counter-clockwise spirals on a sunflower head always form consecutive Fibonacci numbers (e.g., 34 and 55, or 55 and 89). Drawing \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e377\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e377\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e377\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e circles (the 14th Fibonacci number) honors this natural symmetry.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eConstant Packing Density\u003c/strong\u003e: By using the cumulative area to determine radial coordinates, the seed density remains constant from the core to the edges, creating a highly balanced layout.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry modifying these parameters for interesting visual variations:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eAdjust the Seed Count\u003c/strong\u003e:\nChange \u003ccode\u003e\\nbrcircles {377}\u003c/code\u003e to other Fibonacci numbers like \u003ccode\u003e144\u003c/code\u003e (sparser) or \u003ccode\u003e987\u003c/code\u003e (very dense).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eApply Color Gradients\u003c/strong\u003e:\nInstead of solid black (\u003ccode\u003e\\fill[]\u003c/code\u003e), color each circle based on its index \u003ccode\u003e\\b\u003c/code\u003e:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emycolor\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHsb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\b*\u003c/span\u003e0.8, 0.8, 0.9\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[mycolor]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\angle\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\pgfmathresult\u003c/span\u003e) circle [radius=\u003cspan class=\"k\"\u003e\\smRadius\u003c/span\u003e];\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eChange the Scaling Factor (\u003ccode\u003e\\fudge\u003c/code\u003e)\u003c/strong\u003e:\nAdjust \u003ccode\u003e\\fudge {.62}\u003c/code\u003e to \u003ccode\u003e0.8\u003c/code\u003e to overlap the circles into a tight mosaic, or down to \u003ccode\u003e0.4\u003c/code\u003e to introduce open space between them.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eDavid Convent\u0026rsquo;s TikZ code demonstrates how simple mathematical rules can model complex organic structures. Through the Golden Angle and cumulative area calculations, LaTeX allows us to create beautiful botanical designs with absolute precision.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eDavid Convent, \u003cem\u003eSunflower pattern (Phyllotaxy)\u003c/em\u003e, \u003ca href=\"https://texample.net/phyllotaxy/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eJim Bumgardner, \u003cem\u003eCircles, Spirals and Sunflowers\u003c/em\u003e, \u003ca href=\"http://krazydad.com/tutorials/circles_js/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ekrazydad.com\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003cli\u003eStackExchange discussion, \u003cem\u003eDimension too large error with TikZ mod()\u003c/em\u003e, \u003ca href=\"http://tex.stackexchange.com/questions/192028/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003etex.stackexchange.com/questions/192028/\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Drawing an Artistic Poppy Flower using the TikZ Package in LaTeX", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/poppy/", "url" : "https://www.aantriono.com/en/poppy/", "summary": "This article explores vector illustration techniques for an artistic Poppy Flower using the TikZ package in LaTeX, featuring Bézier curves, color gradients, fadings, and clipping techniques.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores vector illustration techniques for an artistic Poppy Flower using the TikZ package in LaTeX, featuring Bézier curves, color gradients, fadings, and clipping techniques.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThe \u003cstrong\u003eTikZ\u003c/strong\u003e package in LaTeX is not limited to formal scientific diagrams, function plots, or rigid mathematical schemes. More than that, TikZ serves as a versatile digital medium for creating beautiful, organic botanical illustrations.\u003c/p\u003e\n\u003cp\u003eA prime example of this artistic capability is the \u003cstrong\u003ePoppy Flower\u003c/strong\u003e (\u003cem\u003ePapaver rhoeas\u003c/em\u003e) illustration created by \u003cstrong\u003eElena Botoeva\u003c/strong\u003e. Combining smooth Bézier curves, rich custom color palettes, and multi-layered lighting effects via \u003cem\u003efadings\u003c/em\u003e and \u003cem\u003eclipping\u003c/em\u003e, Elena achieves a lifelike, dynamic botanical artwork purely using LaTeX code.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code to generate three variations of the poppy flower (full shaded, uniform color, and black-and-white):\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Poppy flower\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Elena Botoeva\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=0.3cm, paperwidth=9.5cm, paperheight=9.5cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efadings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Define Custom Poppy Color Palette\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edarkgreen\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.2,0.6,0.2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epoppyred\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.89,0.08,0.3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epoppyredd\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.78,0.0,0.09\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epoppyreddd\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.63,0.0,0.0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Define Petal Shapes using Bézier Curves\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\petala\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [rounded corners=3] (0,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-0.25,-0.5) and (-1.25,0.3) .. (-1.6,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-1.6,0.75) and (-1.25,0.65) .. (-1.1,1)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-1.05,0.65) and (-0.05,0.7) .. (0,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petalb\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [rounded corners=3] (0,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-0.55,-0.2) and (-1.25,0.7) .. (-1.5,0.6)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-1.4,1.1) and (-1.15,0.9) .. (-0.75,1.4)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-0.5,1.5) and (-0.05,1.45) .. (0.05,1.3)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-0.15,1.1) and (0.25,0.05) .. (0,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petalc\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e[rotate=-30] \u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petald\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e[rotate=-90] \u003cspan class=\"k\"\u003e\\petala\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Define Petiole / Stem Shapes\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\petiolea\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [rounded corners=2] (0.25,-7)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.7,-5.5) and (0.3,-3) .. (0.25,-2)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.2,-1.5) and (0.33,-0.4) .. (0,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.34,-0.4) and (0.21,-1.5) .. (0.27,-2)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.4,-3) and (0.82,-5.5) .. (0.42,-7)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petioleb\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [rounded corners=2] (0.1,-7)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.3,-3) and (-0.85,-1.05) .. (-2.3,-1)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e arc (270:230:0.2)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-2.7,-1) and (-3.05,-1.05) .. (-3.35,-1.3)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-3.35,-1.5) and (-3.45,-1.75) .. (-3.65,-1.75)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-3.7,-1.25) and (-3.4,-1.35) .. (-3.4,-1.25)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-3.05,-0.95) and (-2.7,-0.9) .. (-2.4,-0.85)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-1.1,-0.75) and (0.4,-2.45) .. (0.2,-7)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petiolec\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [rounded corners=2] (-0.3,-7)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-0.9,-3.5) and (-1.5,-2.7) .. (-2,-2.5)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-2.5,-2.3) and (-3.2,-2.35) .. (-3.47,-3.5)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-3.68,-3.85) and (-3.75,-4.2) .. (-3.9,-4.15)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-4.0,-3.9) and (-3.8,-3.8) .. (-3.7,-3.8)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-3.65,-3.75) and (-3.6,-3.7) .. (-3.5,-3.47)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-3.2,-2.3) and (-2.5,-2.25) .. (-2,-2.45)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-1.48,-2.65) and (-0.86,-3.5) .. (-0.26,-7)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Transparency / Fading Configurations\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzfading\u003c/span\u003e\u003cspan class=\"na\"\u003e[name=fade inside, inner color=transparent!100, outer color=transparent!30]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzfading\u003c/span\u003e\u003cspan class=\"na\"\u003e[name=fade out, inner color=transparent!0, outer color=transparent!100]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Shaded Flower Definition with Overlapping Shadows\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\flower\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 1. Draw Each Petal Independently\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\petala\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petald\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=none,color=poppyred,opacity=0.8]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [postaction=\u003cspan class=\"nb\"\u003e{\u003c/span\u003epath fading=south,fading angle=30,fill=poppyreddd,opacity=.2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e] \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 2. Draw All Petals Combined\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=none,rounded corners=3]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [postaction=\u003cspan class=\"nb\"\u003e{\u003c/span\u003epath fading=north,fading angle=30,fill=poppyreddd,opacity=.7\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\petala\u003c/span\u003e \u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e \u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e [rotate=30] \u003cspan class=\"k\"\u003e\\petald\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 3. Emphasize Overlapping Areas Between 2 Petals (Clipping)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\b\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\petala\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\petald\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [draw=none]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [postaction=\u003cspan class=\"nb\"\u003e{\u003c/span\u003epath fading=fade inside,fill=poppyredd,opacity=.7\u003cspan class=\"nb\"\u003e}\u003c/span\u003e] \u003cspan class=\"k\"\u003e\\b\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 4. Emphasize Overlapping Areas Between 3 Petals\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\b\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\petala\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\petald\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e \u003cspan class=\"k\"\u003e\\b\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [draw=none]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [postaction=\u003cspan class=\"nb\"\u003e{\u003c/span\u003epath fading=fade out,fill=poppyreddd,opacity=.4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e] \u003cspan class=\"k\"\u003e\\c\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petioles\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\petiolea\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petioleb\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petiolec\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill,color=#1]\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Full Shaded Poppy Command\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\poppy\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1,#2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,scale=#2]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\flower\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\petioles\u003c/span\u003e\u003cspan class=\"na\"\u003e[darkgreen]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Uniform Color Poppy Command\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\poppyu\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1,#2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,scale=#2]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\petala\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petald\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=none,color=poppyred,opacity=0.8]\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\petioles\u003c/span\u003e\u003cspan class=\"na\"\u003e[darkgreen]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Black and White Poppy Command\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\poppybw\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1,#2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,scale=#2]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\petala\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalb\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petalc\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\petald\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=none,color=gray!80,opacity=1]\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\petioles\u003c/span\u003e\u003cspan class=\"na\"\u003e[gray!80]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\thispagestyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eempty\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\tikz\\poppy\u003c/span\u003e\u003cspan class=\"na\"\u003e[{(0cm,0cm)},1]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\tikz\\poppyu\u003c/span\u003e\u003cspan class=\"na\"\u003e[{(4cm,0cm)},1]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\tikz\\poppybw\u003c/span\u003e\u003cspan class=\"na\"\u003e[{(7cm,0cm)},0.8]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"output-produced\" class=\"headerLink\"\u003e\n \u003ca href=\"#output-produced\" class=\"header-mark\" aria-label=\"Direct link to section: Output Produced\"\u003e\u003c/a\u003eOutput Produced\u003c/h2\u003e\u003cp\u003eThe compiled graphical output displays three elegant variations of the poppy flower:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/poppy/poppy.webp\" title=\"Poppy Flower TikZ Output\" data-thumbnail=\"/poppy/poppy.webp\" data-sub-html=\"\u003ch2\u003eCompiled Poppy Flower Output\u003c/h2\u003e\u003cp\u003ePoppy Flower TikZ Output\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/poppy/poppy.webp' srcset=\"/poppy/poppy_hu_be7fc7adad323783.webp 800w, /poppy/poppy_hu_c30eb87a1e900659.webp 1200w, /poppy/poppy_hu_2b9f59d21e8fa379.webp 1600w\" alt=\"Poppy Flower TikZ Output\" height=\"500\" width=\"500\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCompiled Poppy Flower Output\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Analysis\"\u003e\u003c/a\u003eDetailed Code Analysis\u003c/h2\u003e\u003ch3 id=\"1-smooth-bézier-curves--controls-\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-smooth-b%c3%a9zier-curves--controls-\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Smooth Bézier Curves (.. controls ..)\"\u003e\u003c/a\u003e1. Smooth Bézier Curves (\u003ccode\u003e.. controls ..\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eThe organic contours of the petals and stems are generated using cubic Bézier curves:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e(0,0) .. controls (x1,y1) and (x2,y2) .. (x3,y3)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eControl points \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_1, y_1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_2, y_2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e smoothly pull the paths to craft natural petal curves and sweeping stem lines.\u003c/p\u003e\n\u003ch3 id=\"2-transparency--fadings-library\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-transparency--fadings-library\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Transparency \u0026amp; Fadings Library\"\u003e\u003c/a\u003e2. Transparency \u0026amp; Fadings Library\u003c/h3\u003e\u003cp\u003eThe \u003ccode\u003efadings\u003c/code\u003e TikZ library creates gradual color transitions from opaque to transparent:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzfading\u003c/span\u003e\u003cspan class=\"na\"\u003e[name=fade inside, inner color=transparent!100, outer color=transparent!30]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eCombined with \u003ccode\u003epostaction={path fading=...}\u003c/code\u003e, this technique generates natural lighting highlights on petal folds.\u003c/p\u003e\n\u003ch3 id=\"3-multi-layer-clipping-technique\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-multi-layer-clipping-technique\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Multi-Layer Clipping Technique\"\u003e\u003c/a\u003e3. Multi-Layer Clipping Technique\u003c/h3\u003e\u003cp\u003eThe highlight of Elena Botoeva\u0026rsquo;s design is the use of TikZ clipping to create realistic \u003cstrong\u003eoverlapping shadows\u003c/strong\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [draw=none] [postaction=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e...\u003cspan class=\"nb\"\u003e}\u003c/span\u003e] \u003cspan class=\"k\"\u003e\\b\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eCommand \u003ccode\u003e\\clip \\a;\u003c/code\u003e restricts drawing strictly within the boundary of the first petal (\u003ccode\u003e\\petala\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003eCommand \u003ccode\u003e\\draw ... \\b;\u003c/code\u003e then renders the second petal (\u003ccode\u003e\\petalb\u003c/code\u003e) inside that restricted area.\u003c/li\u003e\n\u003cli\u003eAs a result, dark shadow shading is rendered \u003cstrong\u003eonly where two or three petals intersect\u003c/strong\u003e, giving the flower convincing 3D depth.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry these modifications for creative variations:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eCustom Flower Palette\u003c/strong\u003e:\nModify \u003ccode\u003epoppyred\u003c/code\u003e to create roses (\u003ccode\u003ergb{0.9,0.1,0.1}\u003c/code\u003e), lavender blossoms (\u003ccode\u003ergb{0.6,0.3,0.8}\u003c/code\u003e), or sunflowers (\u003ccode\u003ergb{0.95,0.75,0.1}\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eScale \u0026amp; Rotation Adjustments\u003c/strong\u003e:\nUse \u003ccode\u003e\\poppy[{(x,y)}, scale]\u003c/code\u003e with different values to assemble a multi-flower bouquet.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eElena Botoeva\u0026rsquo;s TikZ poppy code is a masterclass in how advanced vector features—such as \u003cstrong\u003eBézier curves, custom color palettes, fadings, and clipping\u003c/strong\u003e—can be combined to create gallery-quality botanical art.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eElena Botoeva, \u003cem\u003ePoppy flower\u003c/em\u003e, \u003ca href=\"https://texample.net/poppy/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" } { "title" : "Drawing Fractal Snowflakes Using Lindenmayer Systems (L-Systems) in LaTeX TikZ", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/snowflakes/", "url" : "https://www.aantriono.com/en/snowflakes/", "summary": "This article explores Lindenmayer Systems (L-Systems) fractal theory and LaTeX TikZ coding techniques to model 6-fold hexagonal ice crystal snowflakes.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores Lindenmayer Systems (L-Systems) fractal theory and LaTeX TikZ coding techniques to model 6-fold hexagonal ice crystal snowflakes.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn nature, snowflakes form as water vapor freezes around atmospheric particles, crystallizing into hexagonal molecular structures. Consequently, nearly all ice snowflakes exhibit \u003cstrong\u003e6-fold radial symmetry\u003c/strong\u003e (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e60^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e rotation).\u003c/p\u003e\n\u003cp\u003eIn computer science and fractal geometry, one of the most elegant methods for modeling self-similar branching structures is the \u003cstrong\u003eLindenmayer System\u003c/strong\u003e (\u003cem\u003eL-System\u003c/em\u003e), introduced by theoretical biologist Aristid Lindenmayer in 1968.\u003c/p\u003e\n\u003cp\u003eThis article reviews a LaTeX TikZ implementation by \u003cstrong\u003eJose Luis Diaz (JLDiaz)\u003c/strong\u003e. By defining L-System string rewrite rules for a single snowflake \u0026ldquo;arm\u0026rdquo; using the \u003ccode\u003elindenmayersystems\u003c/code\u003e library and rotating it 6 times (\u003ccode\u003e\\foreach \\a in {0,60,...,300}\u003c/code\u003e), we can render a vast gallery of precise, intricate crystal snowflakes.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"mathematical-foundations-lindenmayer-systems-l-systems\" class=\"headerLink\"\u003e\n \u003ca href=\"#mathematical-foundations-lindenmayer-systems-l-systems\" class=\"header-mark\" aria-label=\"Direct link to section: Mathematical Foundations: Lindenmayer Systems (L-Systems)\"\u003e\u003c/a\u003eMathematical Foundations: Lindenmayer Systems (L-Systems)\u003c/h2\u003e\u003cp\u003eA formal L-System is defined as a quintuple \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003eΣ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eω\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eG = (V, \\Sigma, \\omega, P)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003eΣ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eω\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eV\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (Variable Alphabet)\u003c/strong\u003e: Symbols replaced during string rewriting (e.g., \u003ccode\u003eF\u003c/code\u003e, \u003ccode\u003eG\u003c/code\u003e, \u003ccode\u003eH\u003c/code\u003e). In TikZ graphics, \u003ccode\u003eF\u003c/code\u003e commands the turtle/pen to move forward and draw a line segment of length \u003ccode\u003estep\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003eΣ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\Sigma\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003eΣ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (Constant Alphabet)\u003c/strong\u003e: Turtle control symbols:\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e+\u003c/code\u003e : Turn left by \u003ccode\u003eangle\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e-\u003c/code\u003e : Turn right by \u003ccode\u003eangle\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e[\u003c/code\u003e : Push current position and heading onto the state stack.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e]\u003c/code\u003e : Pop position and heading from the stack (used for branching).\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eω\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\omega\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eω\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (Axiom / Initial State)\u003c/strong\u003e: The starting string prior to recursion (e.g., \u003ccode\u003eaxiom=F\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (Production Rules)\u003c/strong\u003e: Rewrite rules replacing variables with new strings at each recursion level (\u003ccode\u003eorder\u003c/code\u003e).\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete TikZ code by Jose Luis Diaz to compute multiple snowflake variations:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Snowflakes with TikZ\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Jose Luis Diaz (JLDiaz)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[textwidth=6cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[active,tightpage]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\PreviewEnvironment\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\\PreviewBorder\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e10pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003elindenmayersystems\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% L-System Rule Declarations for Snowflake Arms\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelindenmayersystem\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eA\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\rule\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eF -\u0026gt; FF[+F][-F]\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelindenmayersystem\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eB\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\rule\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eF -\u0026gt; ffF[++FF][--FF]\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelindenmayersystem\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eC\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\symbol\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eG\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgflsystemdrawforward\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\rule\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eF -\u0026gt; F[+F][-F]FG[+F][-F]FG\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelindenmayersystem\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eD\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\symbol\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eG\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgflsystemdrawforward\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\symbol\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eH\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgflsystemdrawforward\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\rule\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eF -\u0026gt; H[+HG][-HG]G\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\rule\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eG -\u0026gt; HF\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Default Style Configuration\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e type/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1, axiom=F, order=3, step=4pt, angle=60\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e blue, opacity=0.4, line width=.5mm, line cap=round\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Macro to Draw a Complete Snowflake (6-Fold Radial Symmetry)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcommand\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[2][scale=0.2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1]\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,60,...,300\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[rotate=\\a,#2]\u003c/span\u003e l-system;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% System A Variations Across Line Widths\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\width\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.2,.4,...,.8\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.3]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=A, line width=\u003cspan class=\"k\"\u003e\\width\u003c/span\u003e mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% System A Variations with 90-Degree Branch Angle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\width\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.2,.4,...,.8\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.38]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=A, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eangle=90\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=\u003cspan class=\"k\"\u003e\\width\u003c/span\u003e mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% System B Variations\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\width\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.2,.4,...,.8\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.3]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=B, line width=\u003cspan class=\"k\"\u003e\\width\u003c/span\u003e mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\width\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e.2,.4,...,.8\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=B, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eangle=30\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=\u003cspan class=\"k\"\u003e\\width\u003c/span\u003e mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Systems C \u0026amp; D with Custom Axioms\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.24]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=C, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.25]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=C, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.4mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.25]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=C, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=2,axiom=fF\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.32]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=C, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=2,axiom=---fff+++F\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.38]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=D, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=4,angle=60,axiom=GF\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.7mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.38]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=D, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=4,angle=60,axiom=GfF\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.7mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.38]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=D, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=4,angle=60,axiom=FG\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.7mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.38]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etype=D, l-system=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eorder=4,angle=60,axiom=FfG\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, line width=0.7mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"output-produced\" class=\"headerLink\"\u003e\n \u003ca href=\"#output-produced\" class=\"header-mark\" aria-label=\"Direct link to section: Output Produced\"\u003e\u003c/a\u003eOutput Produced\u003c/h2\u003e\u003cp\u003eCompiling the code generates a diverse gallery of snowflake crystals:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/snowflakes/snowflakes.webp\" title=\"Snowflakes TikZ\" data-thumbnail=\"/snowflakes/snowflakes.webp\" data-sub-html=\"\u003ch2\u003eSnowflakes L-Systems TikZ Output\u003c/h2\u003e\u003cp\u003eSnowflakes TikZ\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/snowflakes/snowflakes.webp' srcset=\"/snowflakes/snowflakes_hu_34876130074c1d02.webp 800w, /snowflakes/snowflakes_hu_8e18b47444702acc.webp 1200w, /snowflakes/snowflakes_hu_e4ab6626d41dccd4.webp 1600w\" alt=\"Snowflakes TikZ\" height=\"500\" width=\"417\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSnowflakes L-Systems TikZ Output\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-analysis--logic\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-analysis--logic\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Analysis \u0026amp; Logic\"\u003e\u003c/a\u003eDetailed Code Analysis \u0026amp; Logic\u003c/h2\u003e\u003ch3 id=\"1-lindenmayersystems-library\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-lindenmayersystems-library\" class=\"header-mark\" aria-label=\"Direct link to section: 1. lindenmayersystems Library\"\u003e\u003c/a\u003e1. \u003ccode\u003elindenmayersystems\u003c/code\u003e Library\u003c/h3\u003e\u003cp\u003eTikZ\u0026rsquo;s built-in \u003ccode\u003e\\usetikzlibrary{lindenmayersystems}\u003c/code\u003e provides native string-rewriting capabilities without requiring custom recursive TeX routines.\u003c/p\u003e\n\u003ch3 id=\"2-production-rule-declarations\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-production-rule-declarations\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Production Rule Declarations\"\u003e\u003c/a\u003e2. Production Rule Declarations\u003c/h3\u003e\u003cp\u003eLooking at System A:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelindenmayersystem\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eA\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\rule\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eF -\u0026gt; FF[+F][-F]\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eAt \u003ccode\u003eorder=1\u003c/code\u003e, \u003ccode\u003eF\u003c/code\u003e becomes \u003ccode\u003eFF[+F][-F]\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eAt \u003ccode\u003eorder=2\u003c/code\u003e, each \u003ccode\u003eF\u003c/code\u003e in the expanded string is recursively rewritten again.\u003c/li\u003e\n\u003cli\u003eSquare brackets \u003ccode\u003e[+F]\u003c/code\u003e and \u003ccode\u003e[-F]\u003c/code\u003e branch off to the left and right, then restore the turtle position back to the stem.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-6-fold-radial-symmetry-rotation\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-6-fold-radial-symmetry-rotation\" class=\"header-mark\" aria-label=\"Direct link to section: 3. 6-Fold Radial Symmetry Rotation\"\u003e\u003c/a\u003e3. 6-Fold Radial Symmetry Rotation\u003c/h3\u003e\u003cp\u003eThe complete snowflake is assembled inside \u003ccode\u003e\\drawsnowflake\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcommand\\drawsnowflake\u003c/span\u003e\u003cspan class=\"na\"\u003e[2][scale=0.2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1]\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,60,...,300\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[rotate=\\a,#2]\u003c/span\u003e l-system;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eLoop \u003ccode\u003e\\foreach \\a in {0,60,...,300}\u003c/code\u003e duplicates and rotates 1 L-system branch across angles \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e120\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e240\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0^\\circ, 60^\\circ, 120^\\circ, 180^\\circ, 240^\\circ,\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8686em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e300\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e300^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry these modifications to create new snowflake geometries:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eVarying Branch Angles (\u003ccode\u003eangle\u003c/code\u003e)\u003c/strong\u003e:\nSet \u003ccode\u003eangle=45\u003c/code\u003e to generate 8-fold symmetric snowflakes with \u003ccode\u003e\\foreach \\a in {0,45,...,315}\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eRecursion Depth (\u003ccode\u003eorder\u003c/code\u003e)\u003c/strong\u003e:\nIncrease \u003ccode\u003eorder=3\u003c/code\u003e to \u003ccode\u003eorder=4\u003c/code\u003e or \u003ccode\u003eorder=5\u003c/code\u003e for denser, more intricate crystal filigree.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCustom Color Gradients\u003c/strong\u003e:\nChange \u003ccode\u003eblue, opacity=0.4\u003c/code\u003e to \u003ccode\u003ecyan!80!black\u003c/code\u003e or add glow effects for an icy aesthetic.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eJose Luis Diaz\u0026rsquo;s TikZ code demonstrates how Lindenmayer Systems combined with radial rotational symmetry offer an efficient framework for algorithmic botanical and crystalline modeling in LaTeX.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eJose Luis Diaz, \u003cem\u003eSnowflakes with TikZ\u003c/em\u003e, \u003ca href=\"https://texample.net/snowflakes/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003ePrzemyslaw Prusinkiewicz \u0026amp; Aristid Lindenmayer, \u003cem\u003eThe Algorithmic Beauty of Plants\u003c/em\u003e, Springer-Verlag, 1990.\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Drawing Nested Rotated Polygons and Spiral Patterns with TikZ in LaTeX", "date_published" : "2026-07-18T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/rotated-polygons/", "url" : "https://www.aantriono.com/en/rotated-polygons/", "summary": "This article explores how to draw nested rotated polygon spiral patterns (op-art optical illusions) using loops and color gradients in LaTeX TikZ.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores how to draw nested rotated polygon spiral patterns (op-art optical illusions) using loops and color gradients in LaTeX TikZ.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eOne of the most striking visual effects in \u003cstrong\u003efractal geometry\u003c/strong\u003e and optical art (op-art) is the spiral pattern created by recursively nesting polygons.\u003c/p\u003e\n\u003cp\u003eWhen a polygon is progressively rotated by a fraction of an angle and scaled down toward the center, our eyes perceive a deep 3D optical vortex.\u003c/p\u003e\n\u003cp\u003eThis article explores a TikZ implementation by \u003cstrong\u003eDr. rer. nat. Ismael Gutierrez Garcia\u003c/strong\u003e (extending Alain Matthes\u0026rsquo; rotated triangle concept). We will break down how a simple \u003ccode\u003e\\foreach\u003c/code\u003e loop, relative coordinate positioning with \u003ccode\u003epos=.10\u003c/code\u003e, and progressive color density scaling produce mesmerizing hexagon, pentagon, and square spiral structures.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code to generate the three rotated polygon variations:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Rotated polygons\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Dr. rer. nat. Ismael Gutierrez Garcia\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[usenames,dvipsnames,pdftex]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[active,tightpage]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\PreviewEnvironment\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\\PreviewBorder\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eifthen\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% 1. ROTATED HEXAGON (BlueGreen)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e20\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\def\\couleur\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eBlueGreen\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Initial Vertex Position Setup for Hexagon\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e (0,0) coordinate(A)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 120:6cm) coordinate(B)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 60:6cm) coordinate(C)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 0:6cm) coordinate(D)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++(-60:6cm) coordinate(E)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++(240:6cm) coordinate(F);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=\\couleur!\\thedensity]\u003c/span\u003e (A) -- (B) -- (C) --(D) -- (E) -- (F)-- cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% 40-Step Loop for Rotation \u0026amp; Scaling\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,40\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thedensity\u003c/span\u003e+10\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thedensity\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e coordinate(X) at (A)\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (A) -- (B) coordinate[pos=.10](A)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (C) coordinate[pos=.10](B)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (D) coordinate[pos=.10](C)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (E) coordinate[pos=.10](D)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (F) coordinate[pos=.10](E)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (X) coordinate[pos=.10](F);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=\\couleur!\\thedensity]\u003c/span\u003e (A)--(B)--(C)--(D)--(E)--(F)--cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% 2. ROTATED PENTAGON (LimeGreen)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e20\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\def\\couleur\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eLimeGreen\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e (0,0) coordinate(A)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 144:10cm) coordinate(B)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 72:10cm) coordinate(C)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 0:10cm) coordinate(D)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++(-72:10cm) coordinate(E);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=\\couleur!\\thedensity]\u003c/span\u003e (A) -- (B) -- (C) --(D) -- (E) -- cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,40\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thedensity\u003c/span\u003e+10\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thedensity\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e coordinate(X) at (A)\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (A) -- (B) coordinate[pos=.10](A)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (C) coordinate[pos=.10](B)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (D) coordinate[pos=.10](C)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (E) coordinate[pos=.10](D)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (X) coordinate[pos=.10](E);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=\\couleur!\\thedensity]\u003c/span\u003e (A)--(B)--(C)--(D)--(E)--cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% 3. ROTATED SQUARE (OrangeRed)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% -------------------------------------------------------------\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e20\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\def\\couleur\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eOrangeRed\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e (0,0) coordinate(A)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 90:12cm) coordinate(B)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++( 0:12cm) coordinate(C)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ++(-90:12cm) coordinate(D);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=\\couleur!\\thedensity]\u003c/span\u003e (A) -- (B) -- (C) --(D) -- cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,40\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thedensity\u003c/span\u003e+20\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\setcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thedensity\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e coordinate(X) at (A)\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (A) -- (B) coordinate[pos=.10](A)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (C) coordinate[pos=.10](B)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (D) coordinate[pos=.10](C)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (X) coordinate[pos=.10](D);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=\\couleur!\\thedensity]\u003c/span\u003e (A)--(B)--(C)--(D)--cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"output-produced\" class=\"headerLink\"\u003e\n \u003ca href=\"#output-produced\" class=\"header-mark\" aria-label=\"Direct link to section: Output Produced\"\u003e\u003c/a\u003eOutput Produced\u003c/h2\u003e\u003cp\u003eThe resulting graphical output displays three polygon variations with vibrant color depth and rotation:\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/rotated-polygons/rotated-polygons.webp\" title=\"Rotated Polygons TikZ\" data-thumbnail=\"/rotated-polygons/rotated-polygons.webp\" data-sub-html=\"\u003ch2\u003eRotated Polygons TikZ Output\u003c/h2\u003e\u003cp\u003eRotated Polygons TikZ\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/rotated-polygons/rotated-polygons.webp' srcset=\"/rotated-polygons/rotated-polygons_hu_a2f9d0f583d54957.webp 800w, /rotated-polygons/rotated-polygons_hu_14e6f7d8700d3912.webp 1200w, /rotated-polygons/rotated-polygons_hu_5a6670ca82b6b658.webp 1600w\" alt=\"Rotated Polygons TikZ\" height=\"434\" width=\"500\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eRotated Polygons TikZ Output\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-structure-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-structure-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Structure Analysis\"\u003e\u003c/a\u003eDetailed Code Structure Analysis\u003c/h2\u003e\u003ch3 id=\"1-constructing-initial-polygons-via-polar-vectors\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-constructing-initial-polygons-via-polar-vectors\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Constructing Initial Polygons via Polar Vectors\"\u003e\u003c/a\u003e1. Constructing Initial Polygons via Polar Vectors\u003c/h3\u003e\u003cp\u003eVertex coordinates start at the origin \u003ccode\u003e(0,0)\u003c/code\u003e and connect sequentially using polar vector shifts \u003ccode\u003e++(angle:distance)\u003c/code\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eHexagon (6-gon)\u003c/strong\u003e: Uses \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e60^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e angular increments (\u003ccode\u003e120:6cm\u003c/code\u003e, \u003ccode\u003e60:6cm\u003c/code\u003e, \u003ccode\u003e0:6cm\u003c/code\u003e, \u003ccode\u003e-60:6cm\u003c/code\u003e, \u003ccode\u003e240:6cm\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePentagon (5-gon)\u003c/strong\u003e: Uses \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e72\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e72^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e angular increments (\u003ccode\u003e144:10cm\u003c/code\u003e, \u003ccode\u003e72:10cm\u003c/code\u003e, \u003ccode\u003e0:10cm\u003c/code\u003e, \u003ccode\u003e-72:10cm\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSquare (4-gon)\u003c/strong\u003e: Uses \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e angular increments (\u003ccode\u003e90:12cm\u003c/code\u003e, \u003ccode\u003e0:12cm\u003c/code\u003e, \u003ccode\u003e-90:12cm\u003c/code\u003e).\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-vertex-update-algorithm-pos10\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-vertex-update-algorithm-pos10\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Vertex Update Algorithm pos=.10\"\u003e\u003c/a\u003e2. Vertex Update Algorithm \u003ccode\u003epos=.10\u003c/code\u003e\u003c/h3\u003e\u003cp\u003eThe magic behind the automatic rotation and scaling lies in these lines:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e coordinate(X) at (A)\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[coordinate]\u003c/span\u003e (A) -- (B) coordinate[pos=.10](A)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e -- (C) coordinate[pos=.10](B) ...\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eVariable \u003ccode\u003eX\u003c/code\u003e holds the original location of vertex \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e before it is overwritten.\u003c/li\u003e\n\u003cli\u003eOption \u003ccode\u003epos=.10\u003c/code\u003e picks the point located at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e along the edge from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and assigns it as the new position for vertex \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eBecause every vertex shifts \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e toward its neighbor cyclically, connecting the new vertices creates a polygon that is \u003cstrong\u003eboth rotated and shrunk\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-color-density-gradients-density\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-color-density-gradients-density\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Color Density Gradients (density)\"\u003e\u003c/a\u003e3. Color Density Gradients (\u003ccode\u003edensity\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eOn each loop iteration, the \u003ccode\u003edensity\u003c/code\u003e counter increases:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetcounter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edensity\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\thedensity\u003c/span\u003e+10\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=\\couleur!\\thedensity]\u003c/span\u003e (A)--(B)...;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eExpression \u003ccode\u003eBlueGreen!20\u003c/code\u003e fills the outermost polygon with 20% BlueGreen and 80% white.\u003c/li\u003e\n\u003cli\u003eAs iterations approach the center, color saturation increases (\u003ccode\u003e30%\u003c/code\u003e, \u003ccode\u003e40%\u003c/code\u003e, \u003ccode\u003e50%\u003c/code\u003e, \u0026hellip;), yielding a 3D depth gradient effect.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-experimentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation\"\u003e\u003c/a\u003eTikZ Experimentation\u003c/h2\u003e\u003cp\u003eTry these modifications to explore further:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eVarying Rotation Speed (\u003ccode\u003epos\u003c/code\u003e)\u003c/strong\u003e:\nChange \u003ccode\u003epos=.10\u003c/code\u003e to \u003ccode\u003epos=.05\u003c/code\u003e for tighter rotations with more layers, or \u003ccode\u003epos=.20\u003c/code\u003e for sharper twists.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eIncreasing Loop Iterations\u003c/strong\u003e:\nExtend \u003ccode\u003e\\foreach \\x in {1,...,40}\u003c/code\u003e to \u003ccode\u003e1,...,60\u003c/code\u003e to spiral all the way down to the central point.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCustom Color Schemes\u003c/strong\u003e:\nExperiment with other \u003ccode\u003excolor\u003c/code\u003e palettes such as \u003ccode\u003eRoyalBlue\u003c/code\u003e, \u003ccode\u003ePurplishRed\u003c/code\u003e, or \u003ccode\u003eGoldenrod\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eDr. Ismael Gutierrez Garcia\u0026rsquo;s code demonstrates how concise iterative algorithms in TikZ can produce captivating results. By combining position interpolation (\u003ccode\u003epos=.10\u003c/code\u003e) with dynamic color saturation, TikZ transforms basic geometry into stunning fractal art.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eIsmael Gutierrez Garcia, \u003cem\u003eRotated polygons\u003c/em\u003e, \u003ca href=\"https://texample.net/rotated-polygons/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eAlain Matthes, \u003cem\u003eRotated triangle\u003c/em\u003e, \u003ca href=\"https://texample.net/rotated-triangle/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\u003cbr\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Creating Pompeii Mosaic Patterns with LaTeX TikZ", "date_published" : "2025-11-08T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mosaic-from-pompeii/", "url" : "https://www.aantriono.com/en/mosaic-from-pompeii/", "summary": "This article discusses a comprehensive analysis of Daniel Steger\u0026rsquo;s LaTeX TikZ code for generating Pompeii mosaic patterns — covering code structure, visual philosophy, and scientific and artistic relevance.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses a comprehensive analysis of Daniel Steger\u0026rsquo;s LaTeX TikZ code for generating Pompeii mosaic patterns — covering code structure, visual philosophy, and scientific and artistic relevance.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThis work originates from \u003cem\u003eDaniel Steger\u003c/em\u003e, who reinterpreted ancient mosaic patterns from the \u003cstrong\u003eCasa degli Amorini Dorati\u003c/strong\u003e in Pompeii. Using \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, he reconstructed complex geometric forms through the principles of \u003cstrong\u003ecircle intersection and rotational symmetry\u003c/strong\u003e.\nThe purpose of this article is to dissect the code thoroughly: from technical structure, mathematical principles, visual-philosophical interpretation, to scientific and artistic relevance in the modern era.\u003c/p\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eBelow is the complete code for generating Pompeii mosaic patterns using TikZ:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Daniel Steger\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Source: Mosaic from Pompeji\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Casa degli Armorini Dorati, Living room, mosaic\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminimal\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[cap=round]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Colors\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eanglecolor\u003cspan class=\"nb\"\u003e}{\u003c/span\u003egreen!50!black\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebordercolor\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%Configuration\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\alpha\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% degree\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\layer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[scale=5]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\sinTriDiff\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esin(60-\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\cosTriDiff\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1-cos(60-\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\radiusC\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(\u003cspan class=\"k\"\u003e\\cosTriDiff*\\cosTriDiff\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\sinTriDiff*\\sinTriDiff\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\startAng\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e + atan(\u003cspan class=\"k\"\u003e\\sinTriDiff\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\cosTriDiff\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\al\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\alpha*\\layer\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e,...,\u003cspan class=\"k\"\u003e\\al\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\ang\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\startAng\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\xRs\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\radiusC*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\ang\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\yRs\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\radiusC*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\ang\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\radiusLayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\xRs\u003c/span\u003e + sqrt( 1 - \u003cspan class=\"k\"\u003e\\yRs*\\yRs\u003c/span\u003e )\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\angRs\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eacos(\u003cspan class=\"k\"\u003e\\yRs\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\angRss\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eacos(\u003cspan class=\"k\"\u003e\\radiusC*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\ang\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e))\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eanglecolor\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack!\u003cspan class=\"k\"\u003e\\ang\u003c/span\u003e!green\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\step\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2*\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e - 180\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\stop\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e180-2*\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-180, \u003cspan class=\"k\"\u003e\\step\u003c/span\u003e ,..., \u003cspan class=\"k\"\u003e\\stop\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\deltaAng\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=anglecolor,draw=bordercolor]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\radiusLayer\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e arc (-90+\u003cspan class=\"k\"\u003e\\angRs\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\deltaAng\u003c/span\u003e : \u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e-90+\u003cspan class=\"k\"\u003e\\angRss\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\deltaAng\u003c/span\u003e :1)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e arc (\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e+90-\u003cspan class=\"k\"\u003e\\angRss\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\deltaAng\u003c/span\u003e : 2*\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e+90-\u003cspan class=\"k\"\u003e\\angRs\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\deltaAng\u003c/span\u003e :1)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e arc (\u003cspan class=\"k\"\u003e\\deltaAng\u003c/span\u003e+2*\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e : \u003cspan class=\"k\"\u003e\\deltaAng\u003c/span\u003e : \u003cspan class=\"k\"\u003e\\radiusLayer\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\xRs\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\radiusC*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\al\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\startAng\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\yRs\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\radiusC*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\al\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\startAng\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\radiusLayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\xRs\u003c/span\u003e + sqrt( 1 - \u003cspan class=\"k\"\u003e\\yRs*\\yRs\u003c/span\u003e )\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=2, color=bordercolor]\u003c/span\u003e (0,0) circle (.8*\u003cspan class=\"k\"\u003e\\radiusLayer\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\radiusSmall\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e.7*\u003cspan class=\"k\"\u003e\\radiusLayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-60,0,...,240\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=anglecolor]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\radiusSmall\u003c/span\u003e) arc (-180+\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+60: -180+\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e: \u003cspan class=\"k\"\u003e\\radiusSmall\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e arc (0+\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e: -60+\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e: \u003cspan class=\"k\"\u003e\\radiusSmall\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e arc (120+\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e: 60+\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e: \u003cspan class=\"k\"\u003e\\radiusSmall\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0, 4, ..., 360\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=anglecolor]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e:1) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+3:1.05) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+5:1.05) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+2:1) -- cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=anglecolor]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+5:1.05) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+7:1.05) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+4:1.1) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e+2:1.1) -- cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=1, color=bordercolor]\u003c/span\u003e (0,0) circle (1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=1, color=bordercolor]\u003c/span\u003e (0,0) circle (1.1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/mosaic-from-pompeii/mandala.webp\" title=\"mandala\" data-thumbnail=\"/mosaic-from-pompeii/mandala.webp\" data-sub-html=\"\u003ch2\u003eMosaic Pompeii\u003c/h2\u003e\u003cp\u003emandala\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/mosaic-from-pompeii/mandala.webp' srcset=\"/mosaic-from-pompeii/mandala_hu_4677170b0064eec4.webp 800w, /mosaic-from-pompeii/mandala_hu_136c1b8c73f1309e.webp 1200w, /mosaic-from-pompeii/mandala_hu_435a738f25ec0fa7.webp 1600w\" alt=\"mandala\" height=\"367\" width=\"359\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eMosaic Pompeii\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch2 id=\"code-structure-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#code-structure-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Code Structure Explanation\"\u003e\u003c/a\u003eCode Structure Explanation\u003c/h2\u003e\u003ch3 id=\"initialization-and-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#initialization-and-packages\" class=\"header-mark\" aria-label=\"Direct link to section: Initialization and Packages\"\u003e\u003c/a\u003eInitialization and Packages\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminimal\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe code uses the \u003ccode\u003eminimal\u003c/code\u003e class for pure graphics purposes and the \u003ccode\u003etikz\u003c/code\u003e package for drawing vector shapes.\u003c/p\u003e\n\u003ch3 id=\"color-and-parameter-settings\" class=\"headerLink\"\u003e\n \u003ca href=\"#color-and-parameter-settings\" class=\"header-mark\" aria-label=\"Direct link to section: Color and Parameter Settings\"\u003e\u003c/a\u003eColor and Parameter Settings\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eanglecolor\u003cspan class=\"nb\"\u003e}{\u003c/span\u003egreen!50!black\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorlet\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebordercolor\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\alpha\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\layer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003eanglecolor\u003c/code\u003e: dark green gradient color used for rotational effects.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003ebordercolor\u003c/code\u003e: border of each shape in black.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\alpha\u003c/code\u003e: angular interval between elements (5°).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\layer\u003c/code\u003e: number of main layers (5).\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"basic-geometry-calculations\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-geometry-calculations\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Geometry Calculations\"\u003e\u003c/a\u003eBasic Geometry Calculations\u003c/h3\u003e\u003cp\u003eThis section utilizes trigonometry to determine the position and radius of circles resulting from intersections.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\sinTriDiff\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esin(60-\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\cosTriDiff\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1-cos(60-\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\\radiusC\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(\u003cspan class=\"k\"\u003e\\cosTriDiff\u003c/span\u003e\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2 + \u003cspan class=\"k\"\u003e\\sinTriDiff\u003c/span\u003e\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThese values form the foundation for the intersection points between circles that create the hexagonal or flower pattern.\u003c/p\u003e\n\u003ch3 id=\"creating-segments-and-main-pattern\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-segments-and-main-pattern\" class=\"header-mark\" aria-label=\"Direct link to section: Creating Segments and Main Pattern\"\u003e\u003c/a\u003eCreating Segments and Main Pattern\u003c/h3\u003e\u003cp\u003eThe main \u003ccode\u003e\\foreach\u003c/code\u003e loop block repeats rotation every \u003ccode\u003e\\alpha\u003c/code\u003e degrees.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,\u003cspan class=\"k\"\u003e\\alpha\u003c/span\u003e,...,\u003cspan class=\"k\"\u003e\\al\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIn each iteration:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eCenter point coordinates are calculated.\u003c/li\u003e\n\u003cli\u003eIntersection angles (\u003ccode\u003e\\angRs\u003c/code\u003e, \u003ccode\u003e\\angRss\u003c/code\u003e) are obtained using \u003ccode\u003eacos\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eArc patterns are created through three interconnected \u003ccode\u003earc\u003c/code\u003e commands, forming leaf-like or scale-like shapes.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"center-and-outer-decorative-elements\" class=\"headerLink\"\u003e\n \u003ca href=\"#center-and-outer-decorative-elements\" class=\"header-mark\" aria-label=\"Direct link to section: Center and Outer Decorative Elements\"\u003e\u003c/a\u003eCenter and Outer Decorative Elements\u003c/h3\u003e\u003cp\u003eThis section adds a center circle and small petal ornaments around it.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-60,0,...,240\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eForms six symmetric elements, symbolizing the hexagonal harmony characteristic of Roman mosaics.\nWhile the outer circle:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0, 4, ..., 360\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"produces-a-gear-teeth-pattern-symbolizing-order-and-continuity\" class=\"headerLink\"\u003e\n \u003ca href=\"#produces-a-gear-teeth-pattern-symbolizing-order-and-continuity\" class=\"header-mark\" aria-label=\"Direct link to section: Produces a gear teeth pattern, symbolizing order and continuity.\"\u003e\u003c/a\u003eProduces a \u003cem\u003egear teeth\u003c/em\u003e pattern, symbolizing order and continuity.\u003c/h2\u003e\u003ch2 id=\"visual-and-philosophical-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-and-philosophical-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual and Philosophical Analysis\"\u003e\u003c/a\u003eVisual and Philosophical Analysis\u003c/h2\u003e\u003cp\u003eVisually, this pattern forms \u003cstrong\u003eorder born from repetition and circle intersections\u003c/strong\u003e. In the context of classical Pompeii aesthetics:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eCircles represent \u003cstrong\u003eperfection and eternity\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eIntersections symbolize \u003cstrong\u003ethe connection between earthly and divine elements\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eThe green-black gradient provides \u003cstrong\u003econtrast between life and structure\u003c/strong\u003e, giving spatial depth.\nIts philosophy approaches the concept of \u003cstrong\u003emandala\u003c/strong\u003e, but with an ancient Western geometric touch — not Eastern spirituality, but rather \u003cem\u003eaesthetic rationality\u003c/em\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"tikz-variation-experiments\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-variation-experiments\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Variation Experiments\"\u003e\u003c/a\u003eTikZ Variation Experiments\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eParameter\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Effect\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eExample\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\alpha\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eControls pattern density\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\def\\alpha{10}\u003c/code\u003e → sparser shape\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\layer\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNumber of rotation layers\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\def\\layer{8}\u003c/code\u003e → deeper pattern\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003eanglecolor\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eColor scheme\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eReplace with \u003ccode\u003eblue!50!black\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003escale\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eOverall pattern size\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003escale=3\u003c/code\u003e → denser pattern\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eThese experiments demonstrate the flexibility of \u003cstrong\u003eTikZ as a code-based artistic medium\u003c/strong\u003e.\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"you-can-explore-the-following-parameters-to-create-variations\" class=\"headerLink\"\u003e\n \u003ca href=\"#you-can-explore-the-following-parameters-to-create-variations\" class=\"header-mark\" aria-label=\"Direct link to section: You can explore the following parameters to create variations:\"\u003e\u003c/a\u003eYou can explore the following parameters to create variations:\u003c/h2\u003e\u003ch2 id=\"scientific-and-artistic-relevance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scientific-and-artistic-relevance\" class=\"header-mark\" aria-label=\"Direct link to section: Scientific and Artistic Relevance\"\u003e\u003c/a\u003eScientific and Artistic Relevance\u003c/h2\u003e\u003cp\u003eFrom a scientific perspective:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThis code demonstrates \u003cstrong\u003erotational transformation\u003c/strong\u003e, \u003cstrong\u003ecircle intersection\u003c/strong\u003e, and \u003cstrong\u003epolar symmetry\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eIt can be applied in \u003cstrong\u003ecomputational geometry\u003c/strong\u003e, \u003cstrong\u003evector graphics\u003c/strong\u003e, and \u003cstrong\u003evisual mathematics education\u003c/strong\u003e.\nArtistically:\u003c/li\u003e\n\u003cli\u003eIt revives the aesthetics of \u003cstrong\u003eRoman architecture\u003c/strong\u003e in a digital medium.\u003c/li\u003e\n\u003cli\u003eIt serves as a concrete example of how \u003cstrong\u003ecode can become an art canvas\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThrough exploring Daniel Steger\u0026rsquo;s work, we learn that:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u0026ldquo;Code can be a form of art equal to brush and canvas — only it draws with logic and symmetry.\u0026rdquo;\nWith LaTeX TikZ, the relationship between \u003cstrong\u003escience, art, and geometry\u003c/strong\u003e unites in harmony echoing the glory of Pompeii.\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eDaniel Steger, \u003cem\u003eTikZ example: Mosaic from Pompeji\u003c/em\u003e, \u003ca href=\"https://texample.net/tikz/examples/mosaic-from-pompeji/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeXample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTill Tantau, \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, v3.1.10, 2024.\u003c/li\u003e\n\u003cli\u003ePlato, \u003cem\u003eTimaeus\u003c/em\u003e: on geometry and cosmic order.\u003c/li\u003e\n\u003cli\u003eGombrich, E. H. \u003cem\u003eThe Sense of Order: A Study in the Psychology of Decorative Art\u003c/em\u003e. Oxford University Press, 1979.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Drawing a Mandala with the TikZ Package in LaTeX", "date_published" : "2025-11-07T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/drawing-mandalas/", "url" : "https://www.aantriono.com/en/drawing-mandalas/", "summary": "This article analyzes the aesthetics of a TikZ mandala and provides a detailed explanation of its code structure, covering visual philosophy, mathematical analysis, and technical exploration.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article analyzes the aesthetics of a TikZ mandala and provides a detailed explanation of its code structure, covering visual philosophy, mathematical analysis, and technical exploration.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eA \u003cstrong\u003emandala\u003c/strong\u003e is a visual representation of balance, unity, and harmony.\nIn scientific graphics, \u003cstrong\u003eLaTeX\u003c/strong\u003e with the \u003cstrong\u003eTikZ\u003c/strong\u003e package is a powerful tool for producing precise geometric figures, including complex mandala patterns.\nThis article combines two perspectives: an \u003cstrong\u003eaesthetic analysis\u003c/strong\u003e and a \u003cstrong\u003etechnical, line-by-line breakdown\u003c/strong\u003e of the TikZ code that generates the mandala.\nThe goal is to help readers appreciate both the artistic and mathematical logic behind this mesmerizing design.\u003c/p\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz,border=5]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efadings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzfadingfrompicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[name=mandela]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e filled/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e fill=pgftransparent,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e draw=pgftransparent!0,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e line join=round,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e doubled/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e double=pgftransparent,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e double distance=#1,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e doubled/.default=1.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e... (full code as in the original article) ...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzfadingfrompicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclareradialshading\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emandela\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfpointorigin\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e color(0cm)=(yellow);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e color(0.2cm)=(red);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e color(0.4cm)=(purple);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e color(0.6cm)=(blue);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e color(0.8cm)=(cyan);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e color(1cm)=(cyan)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\shade\u003c/span\u003e [shading=mandela, path fading=mandela, fit fading=false]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-6,-6) rectangle (6,6);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/drawing-mandalas/mandala.webp\" title=\"mandala\" data-thumbnail=\"/drawing-mandalas/mandala.webp\" data-sub-html=\"\u003ch2\u003emandala\u003c/h2\u003e\u003cp\u003emandala\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/drawing-mandalas/mandala.webp' srcset=\"/drawing-mandalas/mandala_hu_61c3b7487aecdc64.webp 800w, /drawing-mandalas/mandala_hu_5ed2f0bc269bc680.webp 1200w, /drawing-mandalas/mandala_hu_8c547876977b9ee1.webp 1600w\" alt=\"mandala\" height=\"642\" width=\"728\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003emandala\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch2 id=\"detailed-code-structure-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-structure-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Structure Explanation\"\u003e\u003c/a\u003eDetailed Code Structure Explanation\u003c/h2\u003e\u003ch3 id=\"declaration-and-libraries\" class=\"headerLink\"\u003e\n \u003ca href=\"#declaration-and-libraries\" class=\"header-mark\" aria-label=\"Direct link to section: Declaration and Libraries\"\u003e\u003c/a\u003eDeclaration and Libraries\u003c/h3\u003e\u003cp\u003eThe code begins with the \u003ccode\u003estandalone\u003c/code\u003e class to output a self-contained image.\nThe fadings library enables gradual transparency effects (\u003cem\u003emasking\u003c/em\u003e).\u003c/p\u003e\n\u003ch3 id=\"defining-the-fading-picture\" class=\"headerLink\"\u003e\n \u003ca href=\"#defining-the-fading-picture\" class=\"header-mark\" aria-label=\"Direct link to section: Defining the Fading Picture\"\u003e\u003c/a\u003eDefining the Fading Picture\u003c/h3\u003e\u003cp\u003e\u003ccode\u003e\\begin{tikzfadingfrompicture}[name=mandela]\u003c/code\u003e defines an internal picture used as a \u003cem\u003emask\u003c/em\u003e.\nAll shapes inside this block define the area where the color gradient will be applied.\u003c/p\u003e\n\u003ch3 id=\"general-style-configuration\" class=\"headerLink\"\u003e\n \u003ca href=\"#general-style-configuration\" class=\"header-mark\" aria-label=\"Direct link to section: General Style Configuration\"\u003e\u003c/a\u003eGeneral Style Configuration\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003efilled:\u003c/strong\u003e marks transparent fill areas.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003edoubled:\u003c/strong\u003e draws double transparent lines.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eline join=round:\u003c/strong\u003e smooths curve intersections.\nHere, \u003ccode\u003epgftransparent\u003c/code\u003e controls transparency rather than color.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"first-radial-pattern-16-segments\" class=\"headerLink\"\u003e\n \u003ca href=\"#first-radial-pattern-16-segments\" class=\"header-mark\" aria-label=\"Direct link to section: First Radial Pattern (16 Segments)\"\u003e\u003c/a\u003eFirst Radial Pattern (16 Segments)\u003c/h3\u003e\u003cp\u003eThe first loop \u003ccode\u003e\\foreach \\i in {0,2,...,15}\u003c/code\u003e draws circular clusters every 22.5°, forming a flower-like pattern around the center.\nEach segment includes:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eA main circle of radius 1.\u003c/li\u003e\n\u003cli\u003e20 small circles around radius 1.\u003c/li\u003e\n\u003cli\u003e24 small circles around radius 6/8.\nThis creates layered petals forming the outer mandala ring.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"complex-petal-pattern-odd-iterations\" class=\"headerLink\"\u003e\n \u003ca href=\"#complex-petal-pattern-odd-iterations\" class=\"header-mark\" aria-label=\"Direct link to section: Complex Petal Pattern (Odd Iterations)\"\u003e\u003c/a\u003eComplex Petal Pattern (Odd Iterations)\u003c/h3\u003e\u003cp\u003eThe second block \u003ccode\u003e\\foreach \\i in {1,3,...,15}\u003c/code\u003e draws the main petals:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eCombines \u003ccode\u003earc\u003c/code\u003e and \u003ccode\u003eBézier curves\u003c/code\u003e for smooth shapes.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003erotate\u003c/code\u003e aligns petal direction.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003escale=-1\u003c/code\u003e flips the coordinate system for symmetry.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e(a)\u003c/code\u003e stores temporary coordinates for curve closure.\nNested loops \u003ccode\u003e\\foreach \\j in {0,1,-1,...}\u003c/code\u003e add micro details around each petal for richer texture.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"middle-layers-radius-2-and-94\" class=\"headerLink\"\u003e\n \u003ca href=\"#middle-layers-radius-2-and-94\" class=\"header-mark\" aria-label=\"Direct link to section: Middle Layers (Radius 2 and 9/4)\"\u003e\u003c/a\u003eMiddle Layers (Radius 2 and 9/4)\u003c/h3\u003e\u003cp\u003eUsing evaluate within \u003ccode\u003e\\foreach\u003c/code\u003e, this layer calculates each angle \u003ccode\u003e\\k\u003c/code\u003e dynamically to build intermediate petal structures between inner and outer rings — adding smooth radial transitions.\nConcentric Inner Circles\nThree circles with radii 3/4, 1/2, and 1/4 are drawn, along with small dots around them, giving depth to the mandala core.\u003c/p\u003e\n\u003ch3 id=\"outer-layer-and-additional-texture\" class=\"headerLink\"\u003e\n \u003ca href=\"#outer-layer-and-additional-texture\" class=\"header-mark\" aria-label=\"Direct link to section: Outer Layer and Additional Texture\"\u003e\u003c/a\u003eOuter Layer and Additional Texture\u003c/h3\u003e\u003cp\u003eBlock \u003ccode\u003e\\path [filled, doubled] circle [radius=9/4];\u003c/code\u003e draws the outer boundary.\nA loop of 32 larger dots at radius 6/4 forms a luminous ring of points around the mandala.\u003c/p\u003e\n\u003ch3 id=\"final-details-and-mask-closure\" class=\"headerLink\"\u003e\n \u003ca href=\"#final-details-and-mask-closure\" class=\"header-mark\" aria-label=\"Direct link to section: Final Details and Mask Closure\"\u003e\u003c/a\u003eFinal Details and Mask Closure\u003c/h3\u003e\u003cp\u003eSmaller Bézier paths add textural highlights near the center.\nFinally, \\end{tikzfadingfrompicture} closes the mask definition.\u003c/p\u003e\n\u003ch3 id=\"color-shading-definition\" class=\"headerLink\"\u003e\n \u003ca href=\"#color-shading-definition\" class=\"header-mark\" aria-label=\"Direct link to section: Color Shading Definition\"\u003e\u003c/a\u003eColor Shading Definition\u003c/h3\u003e\u003cp\u003e\u003ccode\u003e\\pgfdeclareradialshading\u003c/code\u003e defines a color gradient from yellow at the center to cyan at the edges — breathing life into the masked geometry.\u003c/p\u003e\n\u003ch3 id=\"applying-the-shading-to-the-mask\" class=\"headerLink\"\u003e\n \u003ca href=\"#applying-the-shading-to-the-mask\" class=\"header-mark\" aria-label=\"Direct link to section: Applying the Shading to the Mask\"\u003e\u003c/a\u003eApplying the Shading to the Mask\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\shade\u003c/span\u003e [shading=mandela, path fading=mandela, fit fading=false]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-6,-6) rectangle (6,6);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis command draws a 12×12 rectangle filled with the \u003ccode\u003emandela\u003c/code\u003e gradient and shaped by the fading mask, revealing the full mandala in radiant color.\u003c/p\u003e\n\u003ch2 id=\"visual-and-philosophical-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-and-philosophical-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual and Philosophical Analysis\"\u003e\u003c/a\u003eVisual and Philosophical Analysis\u003c/h2\u003e\u003cp\u003eThe resulting mandala expresses geometric harmony:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eCenter:\u003c/strong\u003e symbolizes focus and awareness.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMiddle layers:\u003c/strong\u003e represent expanding energy and growth.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eOuter layers:\u003c/strong\u003e reflect the universal symmetry of balance.\nThe gradient from yellow to blue symbolizes a spiritual journey — from awareness (yellow) to tranquility (blue).\nIts radial symmetry reflects both the order of nature and the mathematical beauty of balance.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"tikz-experimentation-and-variations\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-experimentation-and-variations\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Experimentation and Variations\"\u003e\u003c/a\u003eTikZ Experimentation and Variations\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eChange the gradient colors:\u003c/strong\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ecolor(0cm)=(white);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ecolor(0.4cm)=(magenta);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ecolor(1cm)=(black);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003e\u003cstrong\u003eAdd organic randomness:\u003c/strong\u003e use \u003ccode\u003erotate=rand*360\u003c/code\u003e inside some loops.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSoft shadows:\u003c/strong\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n 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group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eshadows.blur\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eevery path/.append style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblur shadow\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAnimate:\u003c/strong\u003e use the \u003ccode\u003eanimate\u003c/code\u003e package to visualize the mandala forming gradually.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"scientific-and-artistic-relevance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scientific-and-artistic-relevance\" class=\"header-mark\" aria-label=\"Direct link to section: Scientific and Artistic Relevance\"\u003e\u003c/a\u003eScientific and Artistic Relevance\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eScientific aspect:\u003c/strong\u003e demonstrates parametric graphics, coordinate transformation, and symmetry in mathematics.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eArtistic aspect:\u003c/strong\u003e showcases how logic and geometry merge into visual poetry.\nTikZ mandalas are useful for:\u003c/li\u003e\n\u003cli\u003eGeometry visualization and research.\u003c/li\u003e\n\u003cli\u003eIllustrating mathematical symmetry.\u003c/li\u003e\n\u003cli\u003eGenerative art inspired by computation.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThis LaTeX code is a shining example of how algorithms can create \u003cem\u003eaesthetics\u003c/em\u003e.\nWith TikZ, mathematics transforms into visual expressions filled with philosophical and artistic depth.\nA deep understanding of its structure opens endless possibilities for computational art and scientific visualization.\u003c/p\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eTantau, T. \u003cem\u003eThe TikZ and PGF Manual\u003c/em\u003e, Version 3.1.10 (2024).\u003c/li\u003e\n\u003cli\u003eWolfram Research, \u003cem\u003eRadial Symmetry in Geometry and Art\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eTriono, Aan. \u003cem\u003eAesthetic Experiments with TikZ: From Fractals to Mandalas\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eSnyder, J. (2022). \u003cem\u003eMathematical Patterns in Sacred Geometry\u003c/em\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e Aan Triono \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Creating a Mandala with LaTeX TikZ", "date_published" : "2025-11-06T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/drawing-mandalas-with-latex/", "url" : "https://www.aantriono.com/en/drawing-mandalas-with-latex/", "summary": "A complete guide to drawing mandalas using LaTeX TikZ — including full .tex source code and detailed line-by-line explanations.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eA complete guide to drawing mandalas using LaTeX TikZ — including full .tex source code and detailed line-by-line explanations.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"this-article-provides-a-complete-explanation-of-how-the-following-latex-code-produces-beautiful-multi-layered-colored-mandalas-using-geometry-transformations-and-shadow-effects-features-from-tikz\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-article-provides-a-complete-explanation-of-how-the-following-latex-code-produces-beautiful-multi-layered-colored-mandalas-using-geometry-transformations-and-shadow-effects-features-from-tikz\" class=\"header-mark\" aria-label=\"Direct link to section: A mandala is a geometric pattern that symbolizes order, balance, and harmony. Typically, mandalas are used in art and meditation as symbols of unity between the microcosm and macrocosm. In the context of computing, we can draw mandalas automatically with high precision using LaTeX with the TikZ package, a very powerful vector graphics library. This article provides a complete explanation of how the following LaTeX code produces beautiful, multi-layered colored mandalas using geometry, transformations, and shadow effects features from TikZ.\"\u003e\u003c/a\u003eA mandala is a geometric pattern that symbolizes order, balance, and harmony. Typically, mandalas are used in art and meditation as symbols of unity between the microcosm and macrocosm.\nIn the context of computing, we can draw mandalas \u003cstrong\u003eautomatically with high precision\u003c/strong\u003e using \u003cstrong\u003eLaTeX\u003c/strong\u003e with the \u003cstrong\u003eTikZ\u003c/strong\u003e package, a very powerful vector graphics library.\nThis article provides a complete explanation of how the following LaTeX code produces beautiful, multi-layered colored mandalas using geometry, transformations, and shadow effects features from TikZ.\u003c/h2\u003e\u003ch2 id=\"complete-latex-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-latex-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete LaTeX Code\"\u003e\u003c/a\u003eComplete LaTeX Code\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=0.3cm, paperwidth=8.4cm, paperheight=8.4cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eshadows\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efadings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzfading\u003c/span\u003e[name=fade out, inner color=transparent!0,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e outer color=transparent!100]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petal\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [rounded corners=0.5]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-1,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-1,0.6) and (-0.07,0.8).. (0,1)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.07,0.8) and (1,0.6).. (1,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.7,-1) and (-0.7,-1).. (-1,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\background\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1,#2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1]\u003c/span\u003e (0,0) circle (3.9);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[#2]\u003c/span\u003e (0,0) circle (1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\center\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e51.4285,102.857,...,360\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=yellow,rotate=\\a,fill=#1]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-0.08,0) -- (0,0.46) -- (0.08,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=yellow,fill=#1]\u003c/span\u003e (0,0) circle (0.1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\u003c/span\u003e \u003cspan class=\"k\"\u003e\\mandala\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\background\u003c/span\u003e\u003cspan class=\"na\"\u003e[red!30!blue!70!black,blue!70!yellow!50]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ysh\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\xs\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\ys\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\af\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\as\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\b\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\w\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\bl\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 3.06/0.72/0.8/25.71425/51.4285/70/100/100/100/100,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 2.34/1/0.9/25.71425/77.14275/6/90/92/80/80,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 1.8/0.8/0.9/51.4285/102.857/0/75/100/60/70,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 1.5/0.6/0.6/25.71425/77.14275/0/55/100/40/60,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 1.1/0.53/0.58/51.4285/102.857/0/40/100/20/50,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 0.8/0.37/0.45/25.71425/77.14275/0/45/85/20/40,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 0.53/0.1/0.24/25.71425/77.14275/0/10/100/0/30,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 0.49/0.22/0.32/51.4285/102.857/0/10/100/0/50\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\af\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\as\u003c/span\u003e,...,360\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[rotate=\u003cspan class=\"k\"\u003e\\a\u003c/span\u003e,shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(0,\u003cspan class=\"k\"\u003e\\ysh\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,xscale=\u003cspan class=\"k\"\u003e\\xs\u003c/span\u003e,yscale=\u003cspan class=\"k\"\u003e\\ys\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=yellow,fill=yellow!\\y!red!\\b!blue!\\w]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [drop shadow=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eshadow xshift=0.5pt, shadow yshift=-0.5pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e 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class=\"k\"\u003e\\ys\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e \u003cspan class=\"k\"\u003e\\petal\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[path fading=fade out,fill=red!\\r!blue!\\bl!black, opacity=0.7]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (0,-0.35) ellipse (1.2 and 0.75);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[path fading=fade out,fill=red!\\r!blue!\\bl!black, opacity=0.3]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (0,-0.2) ellipse (1.2 and 0.4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[path fading=fade out,fill=red!\\r!blue,opacity=0.2]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-0.4,0.6) -- (0,0.9) -- (0.4,0.6);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\center\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pagestyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eempty\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\parindent\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandala\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[magenta!30!,magenta,purple,violet,]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!40!black,violet,purple,pink,]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[lime!90!black,yellow!95!black,orange,red,]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[green!80!black,yellow,pink,purple!90!,red]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalab\u003c/span\u003e[green!50!black, green!60!black, green!70!black!80!, green!50!,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e green!10!, pink!40!, red!40!,];\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalab\u003c/span\u003e[blue!30!black, blue!40!black, blue!60!black, blue!70!black!80!,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e blue!50!, yellow!40!, yellow!70!,];\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"mandala\" class=\"headerLink\"\u003e\n \u003ca href=\"#mandala\" class=\"header-mark\" aria-label=\"Direct link to section: The output Produced: Mandala \"\u003e\u003c/a\u003eThe output Produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/drawing-mandalas-with-latex/mandala.webp\" title=\"mandala\" data-thumbnail=\"/drawing-mandalas-with-latex/mandala.webp\" data-sub-html=\"\u003ch2\u003eMandala\u003c/h2\u003e\u003cp\u003emandala\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/drawing-mandalas-with-latex/mandala.webp' srcset=\"/drawing-mandalas-with-latex/mandala_hu_735b88a5de8b8f88.webp 800w, /drawing-mandalas-with-latex/mandala_hu_b72afd281fefdc9d.webp 1200w, /drawing-mandalas-with-latex/mandala_hu_6f8464cc13adee35.webp 1600w\" alt=\"mandala\" height=\"546\" width=\"556\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eMandala\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/h2\u003e\u003ch2 id=\"basic-document-setup\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-document-setup\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Document Setup\"\u003e\u003c/a\u003eBasic Document Setup\u003c/h2\u003e\u003cp\u003eThe initial part of the code prepares the document type and canvas size.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 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tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=0.3cm, paperwidth=8.4cm, paperheight=8.4cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eshadows\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efadings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation\"\u003e\u003c/a\u003eExplanation\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003ccode\u003egeometry\u003c/code\u003e is used to set margins and paper size (8.4×8.4 cm).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003etikz\u003c/code\u003e is the main package for drawing.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eshadows\u003c/code\u003e and \u003ccode\u003efadings\u003c/code\u003e add shadow effects and soft gradations to visual elements.\nAdditionally, a fading definition is created so that object edges can fade gradually:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzfading\u003c/span\u003e\u003cspan class=\"na\"\u003e[name=fade out, inner color=transparent!0, outer color=transparent!100]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"creating-the-basic-element-flower-petal\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-the-basic-element-flower-petal\" class=\"header-mark\" aria-label=\"Direct link to section: Creating the Basic Element: Flower Petal\"\u003e\u003c/a\u003eCreating the Basic Element: Flower Petal\u003c/h2\u003e\u003cp\u003eThe main visual element of this mandala is a flower petal shape, defined using Bézier curves:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\petal\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e [rounded corners=0.5]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-1,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (-1,0.6) and (-0.07,0.8).. (0,1)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.07,0.8) and (1,0.6).. (1,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e .. controls (0.7,-1) and (-0.7,-1).. (-1,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"explanation-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-1\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation\"\u003e\u003c/a\u003eExplanation\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eThis shape draws a single symmetric petal with rounded tips.\u003c/li\u003e\n\u003cli\u003eControl points determine the curvature of the petal to appear natural and proportional.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"background-and-mandala-center\" class=\"headerLink\"\u003e\n \u003ca href=\"#background-and-mandala-center\" class=\"header-mark\" aria-label=\"Direct link to section: Background and Mandala Center\"\u003e\u003c/a\u003eBackground and Mandala Center\u003c/h2\u003e\u003ch3 id=\"background\" class=\"headerLink\"\u003e\n \u003ca href=\"#background\" class=\"header-mark\" aria-label=\"Direct link to section: Background\"\u003e\u003c/a\u003eBackground\u003c/h3\u003e\u003cp\u003eFormed with two colored circles:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\background\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1,#2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1]\u003c/span\u003e (0,0) circle (3.9);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[#2]\u003c/span\u003e (0,0) circle (1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"explanation-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-2\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation\"\u003e\u003c/a\u003eExplanation\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eThe first color \u003ccode\u003e#1\u003c/code\u003e is the outer circle color.\u003c/li\u003e\n\u003cli\u003eThe second color \u003ccode\u003e#2\u003c/code\u003e is the center color, providing a contrast effect.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"mandala-center\" class=\"headerLink\"\u003e\n \u003ca href=\"#mandala-center\" class=\"header-mark\" aria-label=\"Direct link to section: Mandala Center\"\u003e\u003c/a\u003eMandala Center\u003c/h3\u003e\u003cp\u003eThe center of the mandala is decorated with small stars with gradient colors:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\center\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\a\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e51.4285,102.857,...,360\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=yellow,rotate=\\a,fill=#1]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-0.08,0) -- (0,0.46) -- (0.08,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=yellow,fill=#1]\u003c/span\u003e (0,0) circle (0.1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"explanation-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-3\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation\"\u003e\u003c/a\u003eExplanation\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eThe star is formed from repeating small triangles rotated by several degrees.\u003c/li\u003e\n\u003cli\u003eThe center color is set through parameter \u003ccode\u003e#1\u003c/code\u003e (usually blue or purple).\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"basic-mandala-fixed-colouring\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-mandala-fixed-colouring\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Mandala (Fixed Colouring)\"\u003e\u003c/a\u003eBasic Mandala (Fixed Colouring)\u003c/h2\u003e\u003cp\u003eThe following definition produces a mandala with a fixed color combination:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\u003c/span\u003e \u003cspan class=\"k\"\u003e\\mandala\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\background\u003c/span\u003e\u003cspan class=\"na\"\u003e[red!30!blue!70!black,blue!70!yellow!50]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\center\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"explanation-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-4\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation\"\u003e\u003c/a\u003eExplanation\u003c/h2\u003e\u003cp\u003eThe core part uses \u003ccode\u003e\\foreach\u003c/code\u003e loops to draw layers of petals repeatedly with rotation and position shifts (\u003ccode\u003erotate\u003c/code\u003e, \u003ccode\u003eshift\u003c/code\u003e, \u003ccode\u003exscale\u003c/code\u003e, \u003ccode\u003eyscale\u003c/code\u003e).\nShadow effects are created with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e[drop shadow=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eshadow xshift=0.5pt, shadow yshift=-0.5pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eSoft light effects using:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[path fading=fade out, fill=red!... , opacity=0.7]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"the-result-is-a-mandala-with-rich-color-nuances-and-real-visual-depth\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-result-is-a-mandala-with-rich-color-nuances-and-real-visual-depth\" class=\"header-mark\" aria-label=\"Direct link to section: The result is a mandala with rich color nuances and real visual depth.\"\u003e\u003c/a\u003eThe result is a mandala with rich color nuances and real visual depth.\u003c/h2\u003e\u003ch2 id=\"parametric-mandala-parametric-colouring\" class=\"headerLink\"\u003e\n \u003ca href=\"#parametric-mandala-parametric-colouring\" class=\"header-mark\" aria-label=\"Direct link to section: Parametric Mandala (Parametric Colouring)\"\u003e\u003c/a\u003eParametric Mandala (Parametric Colouring)\u003c/h2\u003e\u003cp\u003eFor more dynamic color variations, two versions of parametric mandalas are provided.\u003c/p\u003e\n\u003ch3 id=\"mandalac--five-color-parameters\" class=\"headerLink\"\u003e\n \u003ca href=\"#mandalac--five-color-parameters\" class=\"header-mark\" aria-label=\"Direct link to section: \\mandalac — Five Color Parameters\"\u003e\u003c/a\u003e\u003ccode\u003e\\mandalac\u003c/code\u003e — Five Color Parameters\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\u003c/span\u003e \u003cspan class=\"k\"\u003e\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1,#2,#3,#4,#5]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e ... \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"parameter-meanings\" class=\"headerLink\"\u003e\n \u003ca href=\"#parameter-meanings\" class=\"header-mark\" aria-label=\"Direct link to section: Parameter Meanings\"\u003e\u003c/a\u003eParameter Meanings\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e#1\u003c/code\u003e outermost layer color\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e#2\u003c/code\u003e third layer color\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e#3\u003c/code\u003e fifth layer color\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e#4\u003c/code\u003e seventh layer color\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e#5\u003c/code\u003e fading effect color (gradation inward)\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"example-usage\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-usage\" class=\"header-mark\" aria-label=\"Direct link to section: Example Usage\"\u003e\u003c/a\u003eExample Usage\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!40!black,violet,purple,pink,]\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"mandalab--eight-color-parameters\" class=\"headerLink\"\u003e\n \u003ca href=\"#mandalab--eight-color-parameters\" class=\"header-mark\" aria-label=\"Direct link to section: \\mandalab — Eight Color Parameters\"\u003e\u003c/a\u003e\u003ccode\u003e\\mandalab\u003c/code\u003e — Eight Color Parameters\u003c/h3\u003e\u003cp\u003eThis version provides more control over each color layer:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\u003c/span\u003e \u003cspan class=\"k\"\u003e\\mandalab\u003c/span\u003e\u003cspan class=\"na\"\u003e[#1,#2,#3,#4,#5,#6,#7,#8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e ... \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"explanation-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-5\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation\"\u003e\u003c/a\u003eExplanation\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eColor layers are arranged gradually from outside to center.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e#8\u003c/code\u003e controls the intensity of the fading effect (gradation).\nWith this macro, users can produce mandalas with very complex color combinations.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"final-results\" class=\"headerLink\"\u003e\n \u003ca href=\"#final-results\" class=\"header-mark\" aria-label=\"Direct link to section: Final Results\"\u003e\u003c/a\u003eFinal Results\u003c/h2\u003e\u003cp\u003eThe final part of the code produces several mandala variations with different color schemes:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandala\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[magenta!30!,magenta,purple,violet,]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalac\u003c/span\u003e\u003cspan class=\"na\"\u003e[lime!90!black,yellow!95!black,orange,red,]\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikz\\mandalab\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!30!black, blue!40!black, blue!60!black, blue!70!black!80!, blue!50!, yellow!40!, yellow!70!,]\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"results\" class=\"headerLink\"\u003e\n \u003ca href=\"#results\" class=\"header-mark\" aria-label=\"Direct link to section: Results\"\u003e\u003c/a\u003eResults\u003c/h2\u003e\u003cp\u003eA collection of mandala patterns displaying:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eDifferences in color layers and gradations,\u003c/li\u003e\n\u003cli\u003eSoft visual textures due to shadows and fading, and\u003c/li\u003e\n\u003cli\u003eGeometric harmony through regular 360° rotation.\nAll these patterns are generated entirely through code, without the aid of external graphic design software.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eDrawing mandalas with LaTeX and TikZ demonstrates that this scientific typesetting system is not limited to text and formulas, but is also capable of creating complex visual artwork.\nBy utilizing:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eBézier curves for organic shapes,\u003c/li\u003e\n\u003cli\u003eTransformations (rotation, scale, translation) to create symmetry, and\u003c/li\u003e\n\u003cli\u003eShadow and fading effects for visual depth,\nusers can create beautiful, precise, and aesthetic mandala patterns, entirely code-based.\nThis approach combines mathematical logic and artistic beauty — making LaTeX not just a scientific writing tool, but also a digital canvas for geometric art.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eTantau, T. (2024). \u003cem\u003eTikZ \u0026amp; PGF Manual, Version 3.1.10 — The Definitive Guide to Graphics in LaTeX\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eBotoeva, E. (2018). \u003cem\u003eThe Beauty of Symmetry: Drawing Mandalas with TikZ\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eOverleaf. \u003cem\u003eTikZ Examples Gallery\u003c/em\u003e. \u003ca href=\"https://www.overleaf.com/gallery/tagged/tikz\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.overleaf.com/gallery/tagged/tikz\u003c/a\u003e\n\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/li\u003e\n\u003c/ol\u003e", "language": "en" }, { "title" : "The Circle Illusion: Symmetry, Geometry, and Visual Distortion", "date_published" : "2025-11-05T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/visual-distortion/", "url" : "https://www.aantriono.com/en/visual-distortion/", "summary": "This article explores the geometric circle illusion created with LaTeX TikZ, where concentric circles and rotated squares interact to produce visual distortion despite perfect mathematical symmetry.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores the geometric circle illusion created with LaTeX TikZ, where concentric circles and rotated squares interact to produce visual distortion despite perfect mathematical symmetry.\u003c/p\u003e\n\u003chr\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cstrong\u003eNote:\u003c/strong\u003e This article was generated with the help of AI technology and has been manually reviewed to ensure accuracy and clarity.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"this-article-explains-how-to-construct-the-pattern-using-latex-tikz-and-explores-how-the-human-visual-system-interprets-these-shapes-creating-a-misleading-sense-of-imbalance-or-curvature\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-article-explains-how-to-construct-the-pattern-using-latex-tikz-and-explores-how-the-human-visual-system-interprets-these-shapes-creating-a-misleading-sense-of-imbalance-or-curvature\" class=\"header-mark\" aria-label=\"Direct link to section: In the realm of visual perception, simple geometry can create complex illusions. One such case is the concentric circle illusion, where repeated circular patterns appear distorted when combined with rotated square elements. This article explains how to construct the pattern using LaTeX TikZ and explores how the human visual system interprets these shapes, creating a misleading sense of imbalance or curvature.\"\u003e\u003c/a\u003eIn the realm of visual perception, \u003cstrong\u003esimple geometry\u003c/strong\u003e can create complex illusions.\nOne such case is the \u003cstrong\u003econcentric circle illusion\u003c/strong\u003e, where repeated circular patterns appear distorted when combined with \u003cstrong\u003erotated square elements\u003c/strong\u003e.\nThis article explains how to construct the pattern using \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e and explores how the \u003cstrong\u003ehuman visual system\u003c/strong\u003e interprets these shapes, creating a misleading sense of imbalance or curvature.\u003c/h2\u003e\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 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1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[border=10pt,tikz]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pagecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egray!5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[x=4cm,y=4cm,every rectangle node/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003edraw,thick,red,rotate=45,minimum width=0.707*4cm,minimum height=0.707*4cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\diameter\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,6\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[gray,thick]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) circle (1/12*\u003cspan class=\"k\"\u003e\\diameter\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[rectangle]\u003c/span\u003e at (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"ilusi\" class=\"headerLink\"\u003e\n \u003ca href=\"#ilusi\" class=\"header-mark\" aria-label=\"Direct link to section: The output produced: visual distortion \"\u003e\u003c/a\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/visual-distortion/ilusi.webp\" title=\"ilusi\" data-thumbnail=\"/visual-distortion/ilusi.webp\" data-sub-html=\"\u003ch2\u003evisual distortion\u003c/h2\u003e\u003cp\u003eilusi\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/visual-distortion/ilusi.webp' srcset=\"/visual-distortion/ilusi_hu_3192456f6d74ef17.webp 800w, /visual-distortion/ilusi_hu_c8a739a201735f85.webp 1200w, /visual-distortion/ilusi_hu_ad7db354e554559a.webp 1600w\" alt=\"ilusi\" height=\"491\" width=\"516\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003evisual distortion\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/h2\u003e\u003ch2 id=\"code-structure-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#code-structure-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Code Structure Explanation\"\u003e\u003c/a\u003eCode Structure Explanation\u003c/h2\u003e\u003ch3 id=\"document-declaration\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-declaration\" class=\"header-mark\" aria-label=\"Direct link to section: Document Declaration\"\u003e\u003c/a\u003e\u003cstrong\u003eDocument Declaration\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[border=10pt,tikz]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"border10pt-adds-a-small-padding-and-tikz-loads-the-tikz-package-for-vector-graphics\" class=\"headerLink\"\u003e\n \u003ca href=\"#border10pt-adds-a-small-padding-and-tikz-loads-the-tikz-package-for-vector-graphics\" class=\"header-mark\" aria-label=\"Direct link to section: The standalone class generates a single image without full document margins. border=10pt adds a small padding, and tikz loads the TikZ package for vector graphics.\"\u003e\u003c/a\u003eThe \u003ccode\u003estandalone\u003c/code\u003e class generates a single image without full document margins.\n\u003ccode\u003eborder=10pt\u003c/code\u003e adds a small padding, and \u003ccode\u003etikz\u003c/code\u003e loads the TikZ package for vector graphics.\u003c/h2\u003e\u003ch3 id=\"background-color\" class=\"headerLink\"\u003e\n \u003ca href=\"#background-color\" class=\"header-mark\" aria-label=\"Direct link to section: Background Color\"\u003e\u003c/a\u003e\u003cstrong\u003eBackground Color\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pagecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egray!5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"sets-a-light-gray-background-to-improve-contrast-and-visual-comfort\" class=\"headerLink\"\u003e\n \u003ca href=\"#sets-a-light-gray-background-to-improve-contrast-and-visual-comfort\" class=\"header-mark\" aria-label=\"Direct link to section: Sets a light gray background to improve contrast and visual comfort.\"\u003e\u003c/a\u003eSets a light gray background to improve contrast and visual comfort.\u003c/h2\u003e\u003ch3 id=\"tikz-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-environment\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Environment\"\u003e\u003c/a\u003e\u003cstrong\u003eTikZ Environment\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[x=4cm,y=4cm,...]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"defines-the-drawing-area-scaling-ensures-each-grid-unit-equals-4-cm-for-consistent-proportions\" class=\"headerLink\"\u003e\n \u003ca href=\"#defines-the-drawing-area-scaling-ensures-each-grid-unit-equals-4-cm-for-consistent-proportions\" class=\"header-mark\" aria-label=\"Direct link to section: Defines the drawing area. Scaling ensures each grid unit equals 4 cm for consistent proportions.\"\u003e\u003c/a\u003eDefines the drawing area. Scaling ensures each grid unit equals 4 cm for consistent proportions.\u003c/h2\u003e\u003ch3 id=\"rectangle-node-style\" class=\"headerLink\"\u003e\n \u003ca href=\"#rectangle-node-style\" class=\"header-mark\" aria-label=\"Direct link to section: Rectangle Node Style\"\u003e\u003c/a\u003e\u003cstrong\u003eRectangle Node Style\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eevery rectangle node/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003edraw,thick,red,rotate=45,minimum width=0.707*4cm,minimum height=0.707*4cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eDefines the appearance of square nodes:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003edraw,thick,red\u003c/code\u003e: Draws red squares with thick borders.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003erotate=45\u003c/code\u003e: Rotates each square by 45°, forming a diamond shape.\u003c/li\u003e\n\u003cli\u003eDimensions (\u003ccode\u003e0.707*4cm\u003c/code\u003e) ensure proper scaling after rotation (\u003ccode\u003e√2/2 × 4 cm\u003c/code\u003e).\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"grid-of-circles\" class=\"headerLink\"\u003e\n \u003ca href=\"#grid-of-circles\" class=\"header-mark\" aria-label=\"Direct link to section: Grid of Circles\"\u003e\u003c/a\u003e\u003cstrong\u003eGrid of Circles\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e ... \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"generates-a-33-grid-of-circle-clusters-centered-at-coordinates-x-y\" class=\"headerLink\"\u003e\n \u003ca href=\"#generates-a-33-grid-of-circle-clusters-centered-at-coordinates-x-y\" class=\"header-mark\" aria-label=\"Direct link to section: Generates a 3×3 grid of circle clusters centered at coordinates (x, y).\"\u003e\u003c/a\u003eGenerates a 3×3 grid of circle clusters centered at coordinates (x, y).\u003c/h2\u003e\u003ch3 id=\"concentric-circles\" class=\"headerLink\"\u003e\n \u003ca href=\"#concentric-circles\" class=\"header-mark\" aria-label=\"Direct link to section: Concentric Circles\"\u003e\u003c/a\u003e\u003cstrong\u003eConcentric Circles\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\diameter\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,6\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[gray,thick]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) circle (1/12*\u003cspan class=\"k\"\u003e\\diameter\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"draws-six-concentric-circles-per-grid-point-producing-a-layered-depth-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#draws-six-concentric-circles-per-grid-point-producing-a-layered-depth-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: Draws six concentric circles per grid point, producing a layered depth illusion.\"\u003e\u003c/a\u003eDraws six concentric circles per grid point, producing a layered depth illusion.\u003c/h2\u003e\u003ch3 id=\"square-nodes\" class=\"headerLink\"\u003e\n \u003ca href=\"#square-nodes\" class=\"header-mark\" aria-label=\"Direct link to section: Square Nodes\"\u003e\u003c/a\u003e\u003cstrong\u003eSquare Nodes\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[rectangle]\u003c/span\u003e at (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"places-a-red-rotated-square-at-each-grid-center-creating-a-visual-conflict-between-radial-circular-and-diagonal-square-orientation--the-essence-of-the-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#places-a-red-rotated-square-at-each-grid-center-creating-a-visual-conflict-between-radial-circular-and-diagonal-square-orientation--the-essence-of-the-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: Places a red rotated square at each grid center, creating a visual conflict between radial (circular) and diagonal (square) orientation — the essence of the illusion.\"\u003e\u003c/a\u003ePlaces a red rotated square at each grid center, creating a visual conflict between radial (circular) and diagonal (square) orientation — the essence of the illusion.\u003c/h2\u003e\u003ch2 id=\"visual-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual Analysis\"\u003e\u003c/a\u003eVisual Analysis\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eVisual Factor\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003ePerceptual Effect\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eDiagonal squares\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCreate tension and apparent skew\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eConcentric circles\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eReinforce radial focus\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eNeutral colors\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMaintain balance between shapes\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eTogether, they form an impression of distortion even though all elements are mathematically aligned.\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"the-human-eye-perceives-competing-directional-cues\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-human-eye-perceives-competing-directional-cues\" class=\"header-mark\" aria-label=\"Direct link to section: The illusion emerges from the interaction between circle symmetry and diagonal orientation. The human eye perceives competing directional cues:\"\u003e\u003c/a\u003eThe illusion emerges from the \u003cstrong\u003einteraction between circle symmetry and diagonal orientation\u003c/strong\u003e.\nThe human eye perceives competing directional cues:\u003c/h2\u003e\u003ch2 id=\"experimental-variations\" class=\"headerLink\"\u003e\n \u003ca href=\"#experimental-variations\" class=\"header-mark\" aria-label=\"Direct link to section: Experimental Variations\"\u003e\u003c/a\u003eExperimental Variations\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eParameter\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eFunction\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Impact\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eNumber of circles (\u003ccode\u003e1,...,6\u003c/code\u003e)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eControls layering\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMore circles amplify depth illusion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eSquare color\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ee.g., \u003ccode\u003eblue\u003c/code\u003e or \u003ccode\u003eorange\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAlters focal emphasis\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eRotation angle\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChange to 30° or 60°\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eShifts perceived distortion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eCircle spacing\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eModify \u003ccode\u003e1/12\u003c/code\u003e to \u003ccode\u003e1/10\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChanges density and overlap\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eExample variation:\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[rectangle,draw=blue,rotate=30]\u003c/span\u003e at (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"scientific-and-artistic-relevance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scientific-and-artistic-relevance\" class=\"header-mark\" aria-label=\"Direct link to section: Scientific and Artistic Relevance\"\u003e\u003c/a\u003eScientific and Artistic Relevance\u003c/h2\u003e\u003cp\u003eThis pattern demonstrates how \u003cstrong\u003egeometry influences perception\u003c/strong\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eCognitive psychology:\u003c/strong\u003e illustrates orientation conflict and visual interpretation.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eGeometric art:\u003c/strong\u003e transforms mathematical symmetry into aesthetic patterns.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eInformation visualization:\u003c/strong\u003e helps understand alignment perception.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDesign:\u003c/strong\u003e used in architecture and optical art for dynamic compositions.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"using-latex-tikz-such-illusions-can-be-precisely-modeled--turning-mathematics-into-visual-art\" class=\"headerLink\"\u003e\n \u003ca href=\"#using-latex-tikz-such-illusions-can-be-precisely-modeled--turning-mathematics-into-visual-art\" class=\"header-mark\" aria-label=\"Direct link to section: The circle illusion shows that perfect symmetry does not guarantee stable perception. By combining concentric geometry and diagonal rotation, our brains fabricate curvature where none exists. Using LaTeX TikZ, such illusions can be precisely modeled — turning mathematics into visual art.\"\u003e\u003c/a\u003eThe circle illusion shows that \u003cstrong\u003eperfect symmetry does not guarantee stable perception\u003c/strong\u003e.\nBy combining \u003cstrong\u003econcentric geometry and diagonal rotation\u003c/strong\u003e, our brains fabricate curvature where none exists.\nUsing \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, such illusions can be precisely modeled — turning mathematics into visual art.\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eGregory, R. L. (1997). \u003cem\u003eEye and Brain: The Psychology of Seeing.\u003c/em\u003e Oxford University Press.\u003c/li\u003e\n\u003cli\u003eCoren, S., \u0026amp; Girgus, J. S. (1978). \u003cem\u003eSeeing is Deceiving: The Psychology of Visual Illusions.\u003c/em\u003e Lawrence Erlbaum.\u003c/li\u003e\n\u003cli\u003eWare, C. (2013). \u003cem\u003eInformation Visualization: Perception for Design.\u003c/em\u003e Morgan Kaufmann.\u003c/li\u003e\n\u003cli\u003ePGF/TikZ Manual, Version 3.1.10 (2023).\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Drawing Sharingan and Rinnegan Eyes with LaTeX", "date_published" : "2025-10-19T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sharingan-and-rinnegan/", "url" : "https://www.aantriono.com/en/sharingan-and-rinnegan/", "summary": "This article discusses how to draw sharingan and rinnegan eyes from Anime Series: Naruto Shippuden using LaTeX, with the main package used being Tikz package.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses how to draw sharingan and rinnegan eyes from \u003cstrong\u003eAnime Series: Naruto Shippuden\u003c/strong\u003e using LaTeX, with the main package used being Tikz package.\u003c/p\u003e\n\u003chr\u003e\n\u003cblockquote\u003e\n \u003cp\u003eIn the \u003cem\u003eNaruto\u003c/em\u003e anime series, there are several types of eye \u003cem\u003ejutsu\u003c/em\u003e. \u003cstrong\u003eRinnegan\u003c/strong\u003e and \u003cstrong\u003eSharingan\u003c/strong\u003e are two of the three most dangerous \u003cem\u003eDōjutsu\u003c/em\u003e or eye techniques. Both are rare \u003cem\u003eKekkei Genkai\u003c/em\u003e that can only be possessed by specific individuals.\u003c/p\u003e\n\u003cp\u003eSharingan is the characteristic \u003cem\u003eDōjutsu\u003c/em\u003e of the Uchiha clan that allows its users to copy various techniques (\u003cem\u003ejutsu\u003c/em\u003e) such as \u003cem\u003eninjutsu\u003c/em\u003e, \u003cem\u003egenjutsu\u003c/em\u003e, and \u003cem\u003etaijutsu\u003c/em\u003e. Meanwhile, \u003cstrong\u003eRinnegan\u003c/strong\u003e appears when the powers of \u003cem\u003eIndra Otsutsuki\u003c/em\u003e and \u003cem\u003eAsura Otsutsuki\u003c/em\u003e are combined. The main power of Rinnegan is the ability to master life and death — it can even extract a person\u0026rsquo;s soul in a matter of seconds.\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"the-main-package-required-is-tikz-along-with-the-fadings-library-for-coloring-we-use-the-definecolor-command\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-main-package-required-is-tikz-along-with-the-fadings-library-for-coloring-we-use-the-definecolor-command\" class=\"header-mark\" aria-label=\"Direct link to section: In this article, we will learn how to draw Sharingan and Rinnegan eyes using LaTeX. The main package required is TikZ along with the fadings library. For coloring, we use the \\definecolor command.\"\u003e\u003c/a\u003eIn this article, we will learn how to draw \u003cstrong\u003eSharingan\u003c/strong\u003e and \u003cstrong\u003eRinnegan\u003c/strong\u003e eyes using \u003cstrong\u003eLaTeX\u003c/strong\u003e.\nThe main package required is \u003ccode\u003eTikZ\u003c/code\u003e along with the \u003ccode\u003efadings\u003c/code\u003e library. For coloring, we use the \u003ccode\u003e\\definecolor\u003c/code\u003e command.\u003c/h2\u003e\u003ch2 id=\"sharingan\" class=\"headerLink\"\u003e\n \u003ca href=\"#sharingan\" class=\"header-mark\" aria-label=\"Direct link to section: Sharingan\"\u003e\u003c/a\u003eSharingan\u003c/h2\u003e\u003cp\u003eThere are several types of Sharingan in the \u003cem\u003eNaruto\u003c/em\u003e series:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eSharingan with Tomoe\u003c/li\u003e\n\u003cli\u003eMangekyō Sharingan\u003c/li\u003e\n\u003cli\u003eEternal Mangekyō Sharingan\nWe will draw the \u003cstrong\u003ebasic Sharingan with three Tomoe\u003c/strong\u003e, belonging to the Uchiha clan.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"steps\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps\" class=\"header-mark\" aria-label=\"Direct link to section: Steps:\"\u003e\u003c/a\u003eSteps:\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eUse the \u003ccode\u003estandalone\u003c/code\u003e document class with \u003ccode\u003etikz\u003c/code\u003e option:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eLoad TikZ library and create fading effect:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efadings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzfading\u003c/span\u003e\u003cspan class=\"na\"\u003e[name=mripate sasuke, inner color=transparent!0, outer color=transparent!40]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eStart drawing in \u003ccode\u003etikzpicture\u003c/code\u003e environment:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eCreate black square as background:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (-4,-4) rectangle (4,4);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eCreate main red circle (eye):\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[red, path fading=mripate sasuke]\u003c/span\u003e (0,0) circle (3);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eAdd inner black circle and eye center:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick]\u003c/span\u003e (0,0) circle (2);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e (0,0) circle (0.3);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eCreate three Tomoe around the eye:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e80,200,320\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[rotate=\\i]\u003c/span\u003e (2,0) ++ (-30:0.2)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e arc (330:90:0.2) arc (90:-60:0.3) to[out=60,in=-30] cycle;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eClose document:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"final-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#final-result\" class=\"header-mark\" aria-label=\"Direct link to section: Final Result\"\u003e\u003c/a\u003eFinal Result\u003c/h2\u003e\u003ch2 id=\"sharingan-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#sharingan-1\" class=\"header-mark\" aria-label=\"Direct link to section: Sharingan Eye \"\u003e\u003c/a\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/sharingan-and-rinnegan/sharingan.webp\" title=\"Sharingan\" data-thumbnail=\"/sharingan-and-rinnegan/sharingan.webp\" data-sub-html=\"\u003ch2\u003eSharingan Eye\u003c/h2\u003e\u003cp\u003eSharingan\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/sharingan-and-rinnegan/sharingan.webp' srcset=\"/sharingan-and-rinnegan/sharingan_hu_50a87927a5319625.webp 800w, /sharingan-and-rinnegan/sharingan_hu_6dc4207f492419a1.webp 1200w, /sharingan-and-rinnegan/sharingan_hu_bb6686cffa94b831.webp 1600w\" alt=\"Sharingan\" height=\"630\" width=\"630\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSharingan Eye\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/h2\u003e\u003ch2 id=\"rinnegan\" class=\"headerLink\"\u003e\n \u003ca href=\"#rinnegan\" class=\"header-mark\" aria-label=\"Direct link to section: Rinnegan\"\u003e\u003c/a\u003eRinnegan\u003c/h2\u003e\u003cp\u003eRinnegan is formed from the combination of \u003cem\u003eIndra\u003c/em\u003e and \u003cem\u003eAsura Otsutsuki\u0026rsquo;s\u003c/em\u003e powers, granting \u003cstrong\u003eSix Paths Power\u003c/strong\u003e — six different abilities that can be used even by the user\u0026rsquo;s other bodies (such as \u003cem\u003ePain\u003c/em\u003e and \u003cem\u003eObito\u003c/em\u003e).\u003c/p\u003e\n\u003ch3 id=\"steps-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-1\" class=\"header-mark\" aria-label=\"Direct link to section: Steps:\"\u003e\u003c/a\u003eSteps:\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eUse the \u003ccode\u003estandalone\u003c/code\u003e document class:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eDefine colors and basic shapes:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003erinnegan\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eC7B7D1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\mripaterinnegan\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-10.3,-4.6) to[out=5,in=180] (0,-4.4)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e to[out=0,in=250,looseness=1.2] (10.6,4)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e to[out=160,in=55,looseness=1.2] cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eCreate main drawing in \u003ccode\u003etikzpicture\u003c/code\u003e:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eDraw Rinnegan eye pattern and center circle:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[rinnegan]\u003c/span\u003e\u003cspan class=\"k\"\u003e\\mripaterinnegan\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e (0,0) circle (0.4);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eAdd circular lines inside the eye:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\\mripaterinnegan\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1.3,2.7,4.1,5.8,7.3,8.9,10.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) circle (\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[gray,fill opacity=0.8,shift={(-0.4,-0.5)},even odd rule]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-12,-6) rectangle (12,6) \u003cspan class=\"k\"\u003e\\mripaterinnegan\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eDarken the area outside the eye:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[even odd rule]\u003c/span\u003e (-12,-10) rectangle (12,8) \u003cspan class=\"k\"\u003e\\mripaterinnegan\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003eClose document:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"final-result-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#final-result-1\" class=\"header-mark\" aria-label=\"Direct link to section: Final Result\"\u003e\u003c/a\u003eFinal Result\u003c/h2\u003e\u003ch2 id=\"rinnegan-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#rinnegan-1\" class=\"header-mark\" aria-label=\"Direct link to section: Rinnegan Eye \"\u003e\u003c/a\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/sharingan-and-rinnegan/rinnegan.webp\" title=\"Rinnegan\" data-thumbnail=\"/sharingan-and-rinnegan/rinnegan.webp\" data-sub-html=\"\u003ch2\u003eRinnegan Eye\u003c/h2\u003e\u003cp\u003eRinnegan\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/sharingan-and-rinnegan/rinnegan.webp' srcset=\"/sharingan-and-rinnegan/rinnegan_hu_92ca39576353edb4.webp 800w, /sharingan-and-rinnegan/rinnegan_hu_27e901bb01e5ca0e.webp 1200w, /sharingan-and-rinnegan/rinnegan_hu_c3bf0df30af4be7a.webp 1600w\" alt=\"Rinnegan\" height=\"1418\" width=\"1890\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eRinnegan Eye\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/h2\u003e\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"this-process-demonstrates-how-coordinates-points-lines-and-planes-work-together-to-form-precise-and-aesthetic-images--in-accordance-with-the-principles-of-geometry-and-tex-logic-interesting-isnt-it-drawing-with-latex-x1f604\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-process-demonstrates-how-coordinates-points-lines-and-planes-work-together-to-form-precise-and-aesthetic-images--in-accordance-with-the-principles-of-geometry-and-tex-logic-interesting-isnt-it-drawing-with-latex-x1f604\" class=\"header-mark\" aria-label=\"Direct link to section: Through this exercise, we learned to draw unique symbols from the Naruto world mathematically using LaTeX and TikZ. This process demonstrates how coordinates, points, lines, and planes work together to form precise and aesthetic images — in accordance with the principles of geometry and TeX logic. Interesting, isn\u0026rsquo;t it, drawing with LaTeX. 😄\"\u003e\u003c/a\u003eThrough this exercise, we learned to draw unique symbols from the \u003cem\u003eNaruto\u003c/em\u003e world mathematically using \u003cstrong\u003eLaTeX\u003c/strong\u003e and \u003cstrong\u003eTikZ\u003c/strong\u003e.\nThis process demonstrates how coordinates, points, lines, and planes work together to form precise and aesthetic images — in accordance with the principles of \u003cem\u003egeometry\u003c/em\u003e and \u003cem\u003eTeX\u003c/em\u003e logic. Interesting, isn\u0026rsquo;t it, drawing with LaTeX. 😄\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\u003cem\u003eDrawing of Sharingan and Rinnegan Eyes.\u003c/em\u003e\n\u003ca href=\"https://tex.stackexchange.com/questions/680546/drawing-of-sharingan-and-rinnegan-eyes\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://tex.stackexchange.com/questions/680546/drawing-of-sharingan-and-rinnegan-eyes\u003c/a\u003e\n(Accessed on March 24, 2023)\u003c/li\u003e\n\u003cli\u003e\u003cem\u003eSharingan vs Rinnegan: Which One is Stronger.\u003c/em\u003e\n\u003ca href=\"https://www.greenscene.co.id/2020/08/17/naruto-sharingan-vs-rinnegan-mana-yang-lebih-kuat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.greenscene.co.id/2020/08/17/naruto-sharingan-vs-rinnegan-mana-yang-lebih-kuat/\u003c/a\u003e\n(Accessed on March 25, 2023)\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e", "language": "en" }, { "title" : "The Yin-Yang Symbol: The Philosophy of Duality", "date_published" : "2025-11-05T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/yin-yang-symbol/", "url" : "https://www.aantriono.com/en/yin-yang-symbol/", "summary": "This article discusses the philosophy and mathematical structure of the Yin-Yang symbol and demonstrates how to draw it precisely and beautifully using LaTeX TikZ.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the philosophy and mathematical structure of the Yin-Yang symbol and demonstrates how to draw it precisely and beautifully using LaTeX TikZ.\u003c/p\u003e\n\u003chr\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cstrong\u003eNote:\u003c/strong\u003e This article was generated with the help of AI technology and has been manually reviewed to ensure accuracy and clarity.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThe \u003cstrong\u003eYin-Yang symbol (☯)\u003c/strong\u003e is one of the most recognizable icons in Eastern philosophy, particularly within \u003cstrong\u003eTaoism\u003c/strong\u003e.\nIt represents the \u003cstrong\u003eunity of two opposing yet complementary forces\u003c/strong\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eYin\u003c/strong\u003e symbolizes darkness, passivity, femininity, and coldness.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eYang\u003c/strong\u003e symbolizes brightness, activity, masculinity, and warmth.\nThis symbol is not only aesthetically balanced but also a \u003cstrong\u003emathematical visualization of equilibrium\u003c/strong\u003e.\nUsing \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, we can recreate the Yin-Yang symbol with geometric precision and artistic harmony.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eBelow is the complete LaTeX code to draw the Yin-Yang symbol using TikZ:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Color half of the circle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e (0,0) circle (1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0cm,1cm) rectangle (-1cm, -1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Fill the heads\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,0.5) circle (0.5cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (0,-0.5) circle (0.5cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Fill the eyes\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (0,0.5) circle (0.1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,-0.5) circle (0.1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Outer circle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) circle (1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"yinyang\" class=\"headerLink\"\u003e\n \u003ca href=\"#yinyang\" class=\"header-mark\" aria-label=\"Direct link to section: The output produced: yin-yang symbol \"\u003e\u003c/a\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/yin-yang-symbol/yinyang.webp\" title=\"yinyang\" data-thumbnail=\"/yin-yang-symbol/yinyang.webp\" data-sub-html=\"\u003ch2\u003eyin-yang symbol\u003c/h2\u003e\u003cp\u003eyinyang\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/yin-yang-symbol/yinyang.webp' srcset=\"/yin-yang-symbol/yinyang_hu_b852af4189724dba.webp 800w, /yin-yang-symbol/yinyang_hu_852c087d9113c.webp 1200w, /yin-yang-symbol/yinyang_hu_76f9af39808331cc.webp 1600w\" alt=\"yinyang\" height=\"385\" width=\"437\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eyin-yang symbol\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/h2\u003e\u003ch2 id=\"code-structure-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#code-structure-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Code Structure Explanation\"\u003e\u003c/a\u003eCode Structure Explanation\u003c/h2\u003e\u003ch3 id=\"document-declaration\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-declaration\" class=\"header-mark\" aria-label=\"Direct link to section: Document Declaration\"\u003e\u003c/a\u003e\u003cstrong\u003eDocument Declaration\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 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10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 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1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003earticle\u003c/code\u003e defines the basic document type.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003etikz\u003c/code\u003e enables vector-based drawing directly within LaTeX.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"tikz-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-environment\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Environment\"\u003e\u003c/a\u003e\u003cstrong\u003eTikZ Environment\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 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13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e ... \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-environment-contains-all-the-commands-to-draw-the-yin-yang-symbol\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-environment-contains-all-the-commands-to-draw-the-yin-yang-symbol\" class=\"header-mark\" aria-label=\"Direct link to section: This environment contains all the commands to draw the Yin-Yang symbol.\"\u003e\u003c/a\u003eThis environment contains all the commands to draw the Yin-Yang symbol.\u003c/h2\u003e\u003ch3 id=\"drawing-the-half-circle-yin\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-the-half-circle-yin\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing the Half-Circle (Yin)\"\u003e\u003c/a\u003e\u003cstrong\u003eDrawing the Half-Circle (Yin)\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e (0,0) circle (1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0cm,1cm) rectangle (-1cm, -1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\clip\u003c/code\u003e restricts the drawing area to a \u003cstrong\u003ecircle with a 1 cm radius\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eThe rectangle filled with black creates the \u003cstrong\u003eleft half\u003c/strong\u003e of the symbol.\nThe right half remains white, forming the \u003cstrong\u003eYang\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"drawing-the-two-inner-circles-heads-of-yin-and-yang\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-the-two-inner-circles-heads-of-yin-and-yang\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing the Two Inner Circles (Heads of Yin and Yang)\"\u003e\u003c/a\u003e\u003cstrong\u003eDrawing the Two Inner Circles (Heads of Yin and Yang)\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,0.5) circle (0.5cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (0,-0.5) circle (0.5cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eA black circle on top represents \u003cstrong\u003eYang within Yin\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eA white circle on the bottom represents \u003cstrong\u003eYin within Yang\u003c/strong\u003e.\nTogether, they illustrate \u003cstrong\u003ebalance and transformation\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"adding-the-small-circles-eyes\" class=\"headerLink\"\u003e\n \u003ca href=\"#adding-the-small-circles-eyes\" class=\"header-mark\" aria-label=\"Direct link to section: Adding the Small Circles (Eyes)\"\u003e\u003c/a\u003e\u003cstrong\u003eAdding the Small Circles (Eyes)\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (0,0.5) circle (0.1cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,-0.5) circle (0.1cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eThe white dot in the black area and the black dot in the white area symbolize that \u003cstrong\u003eeach element contains the seed of its opposite\u003c/strong\u003e — the core principle of Yin-Yang philosophy.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"drawing-the-outer-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-the-outer-circle\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing the Outer Circle\"\u003e\u003c/a\u003e\u003cstrong\u003eDrawing the Outer Circle\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) circle (1cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"creates-the-boundary-that-unifies-the-entire-shape-and-maintains-visual-harmony\" class=\"headerLink\"\u003e\n \u003ca href=\"#creates-the-boundary-that-unifies-the-entire-shape-and-maintains-visual-harmony\" class=\"header-mark\" aria-label=\"Direct link to section: Creates the boundary that unifies the entire shape and maintains visual harmony.\"\u003e\u003c/a\u003eCreates the boundary that unifies the entire shape and maintains visual harmony.\u003c/h2\u003e\u003ch2 id=\"visual-and-philosophical-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-and-philosophical-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual and Philosophical Analysis\"\u003e\u003c/a\u003eVisual and Philosophical Analysis\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: left\"\u003eElement\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003eColor\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003eSymbolic Meaning\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003eGeometric Form\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eLeft half\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eBlack (Yin)\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eDarkness, night, introspection\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eHalf circle\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eRight half\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eWhite (Yang)\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eLight, day, action\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eHalf circle\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eWhite dot in Yin\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eYang energy in Yin\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eSmall white circle\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eBlack dot in Yang\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eYin energy in Yang\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eSmall black circle\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eThe two sides do not conflict destructively; they \u003cstrong\u003ecomplement each other\u003c/strong\u003e.\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eThis symbol also embodies the idea of \u003cstrong\u003edynamic equilibrium\u003c/strong\u003e — nothing is entirely black or white.\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"experimental-variations\" class=\"headerLink\"\u003e\n \u003ca href=\"#experimental-variations\" class=\"header-mark\" aria-label=\"Direct link to section: Experimental Variations\"\u003e\u003c/a\u003eExperimental Variations\u003c/h2\u003e\u003cp\u003eYou can modify the parameters in TikZ for creative exploration:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eElement\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eModification\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Effect\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eColor\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\fill[blue]\u003c/code\u003e or \u003ccode\u003e\\fill[red]\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChanges symbolic mood\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eSize\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChange \u003ccode\u003e(1cm)\u003c/code\u003e to \u003ccode\u003e(2cm)\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eEnlarges the entire figure\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003ePosition\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdjust \u003ccode\u003e(0,0)\u003c/code\u003e coordinates\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMoves the symbol on the canvas\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eLine style\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdd \u003ccode\u003e[thick, dashed]\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCreates a calligraphic aesthetic\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eExample variation:\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!50!black]\u003c/span\u003e (0,0.5) circle (0.5cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"scientific-and-artistic-relevance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scientific-and-artistic-relevance\" class=\"header-mark\" aria-label=\"Direct link to section: Scientific and Artistic Relevance\"\u003e\u003c/a\u003eScientific and Artistic Relevance\u003c/h2\u003e\u003cp\u003eThe Yin-Yang symbol bridges \u003cstrong\u003ephilosophy, mathematics, and art\u003c/strong\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePhilosophy:\u003c/strong\u003e Represents harmony in opposites.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eGeometry:\u003c/strong\u003e Built from perfect circles and symmetrical relationships.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eArt:\u003c/strong\u003e A timeless visual balance of contrast and unity.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eEducation:\u003c/strong\u003e Ideal for demonstrating geometric symmetry and cultural meaning in LaTeX TikZ.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"every-curve-and-color-embodies-the-harmony-of-science-and-spirituality\" class=\"headerLink\"\u003e\n \u003ca href=\"#every-curve-and-color-embodies-the-harmony-of-science-and-spirituality\" class=\"header-mark\" aria-label=\"Direct link to section: The Yin-Yang symbol beautifully illustrates the unity within duality. Through LaTeX TikZ, we can recreate it precisely while appreciating the deep philosophical essence behind its structure. Every curve and color embodies the harmony of science and spirituality.\"\u003e\u003c/a\u003eThe Yin-Yang symbol beautifully illustrates the \u003cstrong\u003eunity within duality\u003c/strong\u003e.\nThrough \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, we can recreate it precisely while appreciating the deep \u003cstrong\u003ephilosophical essence\u003c/strong\u003e behind its structure.\nEvery curve and color embodies the harmony of science and spirituality.\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eLao Tzu. \u003cem\u003eTao Te Ching\u003c/em\u003e. Trans. D.\nC. Lau. Penguin Classics, 1963.\u003c/li\u003e\n\u003cli\u003eNeedham, J. (1954). \u003cem\u003eScience and Civilization in China\u003c/em\u003e. Cambridge University Press.\u003c/li\u003e\n\u003cli\u003ePGF/TikZ Manual, Version 3.1.10 (2023).\u003c/li\u003e\n\u003cli\u003eLi, X. (2018). \u003cem\u003eYin-Yang Theory and Its Mathematical Interpretation\u003c/em\u003e. Beijing University Press.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "The Café Wall Illusion: Straight Lines That Appear Curved", "date_published" : "2025-10-31T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cafe-wall-illusion/", "url" : "https://www.aantriono.com/en/cafe-wall-illusion/", "summary": "This article explores the Café Wall Illusion, where parallel lines appear curved or slanted due to alternating black and white squares and horizontal offsets. Built using LaTeX TikZ, the article provides a complete explanation of both the code and the perceptual mechanisms behind it.\u0026quot;\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores the Café Wall Illusion, where parallel lines appear curved or slanted due to alternating black and white squares and horizontal offsets. Built using LaTeX TikZ, the article provides a complete explanation of both the code and the perceptual mechanisms behind it.\u0026quot;\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"in-this-article-we-recreate-this-illusion-using-latex-tikz-explaining-both-the-structure-of-the-source-code-and-the-visual-science-behind-the-phenomenon\" class=\"headerLink\"\u003e\n \u003ca href=\"#in-this-article-we-recreate-this-illusion-using-latex-tikz-explaining-both-the-structure-of-the-source-code-and-the-visual-science-behind-the-phenomenon\" class=\"header-mark\" aria-label=\"Direct link to section: Have you ever looked at a tiled wall or brick pattern that seemed wavy even though the lines were perfectly straight? This is known as the Café Wall Illusion, an optical illusion where parallel horizontal lines appear tilted or curved due to the alternating contrast of black and white tiles and horizontal offset between rows. In this article, we recreate this illusion using LaTeX TikZ, explaining both the structure of the source code and the visual science behind the phenomenon.\"\u003e\u003c/a\u003eHave you ever looked at a tiled wall or brick pattern that seemed wavy even though the lines were perfectly straight?\nThis is known as the \u003cstrong\u003eCafé Wall Illusion\u003c/strong\u003e, an optical illusion where \u003cstrong\u003eparallel horizontal lines\u003c/strong\u003e appear tilted or curved due to the \u003cstrong\u003ealternating contrast of black and white tiles\u003c/strong\u003e and \u003cstrong\u003ehorizontal offset between rows\u003c/strong\u003e.\nIn this article, we recreate this illusion using \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, explaining both the \u003cstrong\u003estructure of the source code\u003c/strong\u003e and the \u003cstrong\u003evisual science\u003c/strong\u003e behind the phenomenon.\u003c/h2\u003e\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eBelow is the full LaTeX code used to generate the Café Wall pattern:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pagecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egray\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\offsety\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e.05\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e / \u003cspan class=\"k\"\u003e\\offsetx\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0/0.3,1/0.5,2/0.3,3/0,4/0.3,5/0.5,6/0.3,7/0,8/0.3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,10\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emod(\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,2) == 0\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u0026#34;black\u0026#34;\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u0026#34;white\u0026#34;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\pgfmathresult]\u003c/span\u003e (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"o\"\u003e)+(\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\offsetx\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\offsety\u003c/span\u003e\u003cspan class=\"o\"\u003e*\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e) rectangle +(1,1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"ilusi\" class=\"headerLink\"\u003e\n \u003ca href=\"#ilusi\" class=\"header-mark\" aria-label=\"Direct link to section: The output produced: They are or not parallels? \"\u003e\u003c/a\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/cafe-wall-illusion/ilusi.webp\" title=\"ilusi\" data-thumbnail=\"/cafe-wall-illusion/ilusi.webp\" data-sub-html=\"\u003ch2\u003e They are or not parallels? \u003c/h2\u003e\u003cp\u003eilusi\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/cafe-wall-illusion/ilusi.webp' srcset=\"/cafe-wall-illusion/ilusi_hu_70e9b8df9d47beff.webp 800w, /cafe-wall-illusion/ilusi_hu_b40e02952392dfec.webp 1200w, /cafe-wall-illusion/ilusi_hu_300d5d2c0ef6c783.webp 1600w\" alt=\"ilusi\" height=\"451\" width=\"547\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003e They are or not parallels? \u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/h2\u003e\u003ch2 id=\"explanation-of-code-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-of-code-structure\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation of Code Structure\"\u003e\u003c/a\u003eExplanation of Code Structure\u003c/h2\u003e\u003ch3 id=\"document-declaration\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-declaration\" class=\"header-mark\" aria-label=\"Direct link to section: Document Declaration\"\u003e\u003c/a\u003e\u003cstrong\u003eDocument Declaration\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"the-tikz-option-loads-the-tikz-package-for-creating-vector-graphics-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-tikz-option-loads-the-tikz-package-for-creating-vector-graphics-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: The standalone class generates only the figure without additional margins or headers. The tikz option loads the TikZ package for creating vector graphics in LaTeX.\"\u003e\u003c/a\u003eThe \u003ccode\u003estandalone\u003c/code\u003e class generates only the figure without additional margins or headers.\nThe \u003ccode\u003etikz\u003c/code\u003e option loads the \u003cstrong\u003eTikZ\u003c/strong\u003e package for creating vector graphics in LaTeX.\u003c/h2\u003e\u003ch3 id=\"library-import\" class=\"headerLink\"\u003e\n \u003ca href=\"#library-import\" class=\"header-mark\" aria-label=\"Direct link to section: Library Import\"\u003e\u003c/a\u003e\u003cstrong\u003eLibrary Import\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"the-calc-library-allows-for-dynamic-coordinate-calculations-and-mathematical-positioning-within-tikz\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-calc-library-allows-for-dynamic-coordinate-calculations-and-mathematical-positioning-within-tikz\" class=\"header-mark\" aria-label=\"Direct link to section: The calc library allows for dynamic coordinate calculations and mathematical positioning within TikZ.\"\u003e\u003c/a\u003eThe \u003ccode\u003ecalc\u003c/code\u003e library allows for dynamic coordinate calculations and mathematical positioning within TikZ.\u003c/h2\u003e\u003ch3 id=\"page-background-color\" class=\"headerLink\"\u003e\n \u003ca href=\"#page-background-color\" class=\"header-mark\" aria-label=\"Direct link to section: Page Background Color\"\u003e\u003c/a\u003e\u003cstrong\u003ePage Background Color\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pagecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egray\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"applies-a-gray-background-to-enhance-contrast-between-black-and-white-squares\" class=\"headerLink\"\u003e\n \u003ca href=\"#applies-a-gray-background-to-enhance-contrast-between-black-and-white-squares\" class=\"header-mark\" aria-label=\"Direct link to section: Applies a gray background to enhance contrast between black and white squares.\"\u003e\u003c/a\u003eApplies a gray background to enhance contrast between black and white squares.\u003c/h2\u003e\u003ch3 id=\"drawing-area\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-area\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Area\"\u003e\u003c/a\u003e\u003cstrong\u003eDrawing Area\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"starts-the-tikz-drawing-environment-that-contains-all-visual-elements\" class=\"headerLink\"\u003e\n \u003ca href=\"#starts-the-tikz-drawing-environment-that-contains-all-visual-elements\" class=\"header-mark\" aria-label=\"Direct link to section: Starts the TikZ drawing environment that contains all visual elements.\"\u003e\u003c/a\u003eStarts the TikZ drawing environment that contains all visual elements.\u003c/h2\u003e\u003ch3 id=\"vertical-offset-variable\" class=\"headerLink\"\u003e\n \u003ca href=\"#vertical-offset-variable\" class=\"header-mark\" aria-label=\"Direct link to section: Vertical Offset Variable\"\u003e\u003c/a\u003e\u003cstrong\u003eVertical Offset Variable\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\offsety\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e.05\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"defines-a-small-vertical-offset-of-005-per-row-introducing-subtle-wave-like-distortions-that-enhance-the-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#defines-a-small-vertical-offset-of-005-per-row-introducing-subtle-wave-like-distortions-that-enhance-the-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: Defines a small vertical offset of 0.05 per row, introducing subtle wave-like distortions that enhance the illusion.\"\u003e\u003c/a\u003eDefines a small vertical offset of \u003ccode\u003e0.05\u003c/code\u003e per row, introducing subtle wave-like distortions that enhance the illusion.\u003c/h2\u003e\u003ch3 id=\"row-loop-and-horizontal-offset\" class=\"headerLink\"\u003e\n \u003ca href=\"#row-loop-and-horizontal-offset\" class=\"header-mark\" aria-label=\"Direct link to section: Row Loop and Horizontal Offset\"\u003e\u003c/a\u003e\u003cstrong\u003eRow Loop and Horizontal Offset\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e / \u003cspan class=\"k\"\u003e\\offsetx\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0/0.3,1/0.5,2/0.3,3/0,4/0.3,5/0.5,6/0.3,7/0,8/0.3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eIterates over nine rows (\u003ccode\u003e\\y\u003c/code\u003e from 0 to 8).\u003c/li\u003e\n\u003cli\u003eEach row has a distinct horizontal offset (\u003ccode\u003e\\offsetx\u003c/code\u003e), alternating shifts that make the straight lines appear tilted.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"column-loop-and-square-colors\" class=\"headerLink\"\u003e\n \u003ca href=\"#column-loop-and-square-colors\" class=\"header-mark\" aria-label=\"Direct link to section: Column Loop and Square Colors\"\u003e\u003c/a\u003e\u003cstrong\u003eColumn Loop and Square Colors\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,10\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emod(\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,2) == 0\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u0026#34;black\u0026#34;\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u0026#34;white\u0026#34;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\pgfmathresult]\u003c/span\u003e (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"o\"\u003e)+(\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\offsetx\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\offsety\u003c/span\u003e\u003cspan class=\"o\"\u003e*\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e) rectangle +(1,1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eStep-by-step breakdown:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThe outer loop draws 11 squares per row.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\pgfmathifthenelse{mod(\\x,2) == 0}{\u0026quot;black\u0026quot;}{\u0026quot;white\u0026quot;}\u003c/code\u003e alternates square colors (black for even, white for odd).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\fill[...] ... rectangle +(1,1);\u003c/code\u003e draws 1×1 unit squares at the calculated positions.\u003c/li\u003e\n\u003cli\u003eThe position formula shifts each square horizontally and vertically for illusion enhancement.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"closing-the-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing-the-environment\" class=\"header-mark\" aria-label=\"Direct link to section: Closing the Environment\"\u003e\u003c/a\u003e\u003cstrong\u003eClosing the Environment\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"ends-the-tikz-environment-the-result-is-a-staggered-tile-pattern-that-produces-a-strong-illusion-of-curved-or-slanted-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#ends-the-tikz-environment-the-result-is-a-staggered-tile-pattern-that-produces-a-strong-illusion-of-curved-or-slanted-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Ends the TikZ environment. The result is a staggered tile pattern that produces a strong illusion of curved or slanted lines.\"\u003e\u003c/a\u003eEnds the TikZ environment. The result is a staggered tile pattern that produces a strong illusion of curved or slanted lines.\u003c/h2\u003e\u003ch2 id=\"visual-and-perceptual-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-and-perceptual-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual and Perceptual Analysis\"\u003e\u003c/a\u003eVisual and Perceptual Analysis\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eFactor\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Effect\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eRow offset\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCreates slanted appearance\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eBlack–white contrast\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eIncreases perceptual distortion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eGray background\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdds depth and amplifies curvature illusion\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"although-all-rows-of-tiles-are-mathematically-parallel-our-brain-misinterprets-them-because-of-lateral-inhibition--a-neural-mechanism-where-high-contrast-edges-enhance-perceived-boundaries\" class=\"headerLink\"\u003e\n \u003ca href=\"#although-all-rows-of-tiles-are-mathematically-parallel-our-brain-misinterprets-them-because-of-lateral-inhibition--a-neural-mechanism-where-high-contrast-edges-enhance-perceived-boundaries\" class=\"header-mark\" aria-label=\"Direct link to section: Although all rows of tiles are mathematically parallel, our brain misinterprets them because of lateral inhibition — a neural mechanism where high-contrast edges enhance perceived boundaries.\"\u003e\u003c/a\u003eAlthough all rows of tiles are mathematically parallel, our brain misinterprets them because of \u003cstrong\u003elateral inhibition\u003c/strong\u003e — a neural mechanism where high-contrast edges enhance perceived boundaries.\u003c/h2\u003e\u003ch2 id=\"experiment-and-variation\" class=\"headerLink\"\u003e\n \u003ca href=\"#experiment-and-variation\" class=\"header-mark\" aria-label=\"Direct link to section: Experiment and Variation\"\u003e\u003c/a\u003eExperiment and Variation\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eParameter\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eFunction\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Effect\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\offsetx\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eHorizontal shift between rows\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdjusts illusion strength\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\offsety\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eVertical offset\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eIntroduces wave-like appearance\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eColor palette\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChange square colors\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAlters perceived contrast\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eSquare size\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eModify \u003ccode\u003e(1,1)\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eDenser grid strengthens illusion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eExample variation:\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e / \u003cspan class=\"k\"\u003e\\offsetx\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0/0,1/0.2,2/0.4,3/0.6\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,12\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emod(\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,2)==0\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u0026#34;black\u0026#34;\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u0026#34;white\u0026#34;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\pgfmathresult]\u003c/span\u003e (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"o\"\u003e)+(\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\offsetx\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"nb\"\u003e.\u003c/span\u003e\u003cspan class=\"m\"\u003e05\u003c/span\u003e\u003cspan class=\"o\"\u003e*\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e) rectangle +(1,1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"scientific-and-artistic-relevance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scientific-and-artistic-relevance\" class=\"header-mark\" aria-label=\"Direct link to section: Scientific and Artistic Relevance\"\u003e\u003c/a\u003eScientific and Artistic Relevance\u003c/h2\u003e\u003cp\u003eThe Café Wall Illusion bridges the gap between \u003cstrong\u003evisual psychology\u003c/strong\u003e, \u003cstrong\u003eneuroscience\u003c/strong\u003e, and \u003cstrong\u003emathematical art\u003c/strong\u003e.\nIt provides insight into:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePerceptual psychology\u003c/strong\u003e — understanding how contrast and alignment affect visual interpretation.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eVisual neuroscience\u003c/strong\u003e — studying edge detection and spatial inhibition.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eGraphic design and architecture\u003c/strong\u003e — creating visual depth or motion effects.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMathematical aesthetics\u003c/strong\u003e — blending geometry with perception.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"using-latex-tikz-we-can-mathematically-and-artistically-reconstruct-this-optical-phenomenon-merging-science-art-and-geometry-into-a-single-elegant-visualization\" class=\"headerLink\"\u003e\n \u003ca href=\"#using-latex-tikz-we-can-mathematically-and-artistically-reconstruct-this-optical-phenomenon-merging-science-art-and-geometry-into-a-single-elegant-visualization\" class=\"header-mark\" aria-label=\"Direct link to section: The Café Wall Illusion demonstrates that human perception is context-dependent and contrast-driven. Using LaTeX TikZ, we can mathematically and artistically reconstruct this optical phenomenon, merging science, art, and geometry into a single elegant visualization.\"\u003e\u003c/a\u003eThe Café Wall Illusion demonstrates that human perception is \u003cstrong\u003econtext-dependent\u003c/strong\u003e and \u003cstrong\u003econtrast-driven\u003c/strong\u003e.\nUsing \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, we can mathematically and artistically reconstruct this optical phenomenon, merging \u003cstrong\u003escience, art, and geometry\u003c/strong\u003e into a single elegant visualization.\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eGregory, R. L. (1997). \u003cem\u003eEye and Brain: The Psychology of Seeing.\u003c/em\u003e Oxford University Press.\u003c/li\u003e\n\u003cli\u003eWade, N. J. (1986). \u003cem\u003ePerception and Illusion: Historical Perspectives.\u003c/em\u003e Springer.\u003c/li\u003e\n\u003cli\u003eFraser, J. (1908). \u003cem\u003eA New Visual Illusion of Direction.\u003c/em\u003e British Journal of Psychology.\u003c/li\u003e\n\u003cli\u003ePGF/TikZ Manual, Version 3.1.10 (2023).\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "White Dot Illusion: Perceptual Contrast in a Dark Grid Pattern", "date_published" : "2025-10-02T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/white-dot-illusion/", "url" : "https://www.aantriono.com/en/white-dot-illusion/", "summary": "This article discusses the visual illusion of white dots appearing in a dark grid pattern, where the brain creates contrast perceptions that are not physically present. Built using LaTeX PSTricks, it provides a complete explanation of the code structure and the perceptual mechanisms behind it.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the visual illusion of white dots appearing in a dark grid pattern, where the brain creates contrast perceptions that are not physically present. Built using LaTeX PSTricks, it provides a complete explanation of the code structure and the perceptual mechanisms behind it.\u003c/p\u003e\n\u003chr\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cstrong\u003eNote:\u003c/strong\u003e This article was generated with the help of AI technology and has been manually reviewed to ensure accuracy and clarity.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"in-this-article-we-recreate-this-effect-using-latex-pstricks-and-explain-the-structure-of-the-code-along-with-the-underlying-perceptual-science\" class=\"headerLink\"\u003e\n \u003ca href=\"#in-this-article-we-recreate-this-effect-using-latex-pstricks-and-explain-the-structure-of-the-code-along-with-the-underlying-perceptual-science\" class=\"header-mark\" aria-label=\"Direct link to section: Optical illusions reveal how the human brain interprets visual information beyond mere physical reality. One fascinating example is the Hermann Grid Illusion, also known as the White Dot Illusion, where white or gray dots appear and disappear at the intersections of dark grid lines. This phenomenon occurs due to lateral inhibition in the retina — a neural mechanism that enhances contrast but also leads to perceptual distortions. In this article, we recreate this effect using LaTeX PSTricks and explain the structure of the code along with the underlying perceptual science.\"\u003e\u003c/a\u003eOptical illusions reveal how the human brain interprets visual information beyond mere physical reality. One fascinating example is the \u003cstrong\u003eHermann Grid Illusion\u003c/strong\u003e, also known as the \u003cstrong\u003eWhite Dot Illusion\u003c/strong\u003e, where white or gray dots appear and disappear at the intersections of dark grid lines.\nThis phenomenon occurs due to \u003cstrong\u003elateral inhibition\u003c/strong\u003e in the retina — a neural mechanism that enhances contrast but also leads to perceptual distortions.\nIn this article, we recreate this effect using \u003cstrong\u003eLaTeX PSTricks\u003c/strong\u003e and explain the structure of the code along with the underlying perceptual science.\u003c/h2\u003e\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eBelow is the complete LaTeX source used to visualize the illusion:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[pstricks,border=12pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpsstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egridstyle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e gridlabels=0,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e gridwidth=6pt,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e subgriddiv=1,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e gridcolor=gray,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epspicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e(8,8)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\psframe*\u003c/span\u003e(8,8)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\psgrid\u003c/span\u003e\u003cspan class=\"na\"\u003e[style=gridstyle]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\psset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003elinecolor=white\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\multips\u003c/span\u003e(0,1)(0,1)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e7\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\multips\u003c/span\u003e(1,0)(1,0)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e7\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\qdisk\u003c/span\u003e(0,0)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e4.242pt\u003cspan class=\"nb\"\u003e}}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epspicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/white-dot-illusion/ilusi.webp\" title=\"ilusi titik\" data-thumbnail=\"/white-dot-illusion/ilusi.webp\" data-sub-html=\"\u003ch2\u003eWhite Dot Illusion\u003c/h2\u003e\u003cp\u003eilusi titik\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/white-dot-illusion/ilusi.webp' srcset=\"/white-dot-illusion/ilusi_hu_ca5fb16eedde017c.webp 800w, /white-dot-illusion/ilusi_hu_8f3b8ca01127605.webp 1200w, /white-dot-illusion/ilusi_hu_6f234d9bf3712aca.webp 1600w\" alt=\"ilusi titik\" height=\"682\" width=\"684\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eWhite Dot Illusion\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch2 id=\"explanation-of-code-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-of-code-structure\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation of Code Structure\"\u003e\u003c/a\u003eExplanation of Code Structure\u003c/h2\u003e\u003ch3 id=\"document-declaration\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-declaration\" class=\"header-mark\" aria-label=\"Direct link to section: Document Declaration\"\u003e\u003c/a\u003e\u003cstrong\u003eDocument Declaration\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[pstricks,border=12pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"the-pstricks-option-enables-pstricks-a-latex-package-for-creating-postscript-based-vector-graphics\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-pstricks-option-enables-pstricks-a-latex-package-for-creating-postscript-based-vector-graphics\" class=\"header-mark\" aria-label=\"Direct link to section: Specifies that the document uses the standalone class, which produces a single figure without page margins or headers. The pstricks option enables PSTricks, a LaTeX package for creating PostScript-based vector graphics.\"\u003e\u003c/a\u003eSpecifies that the document uses the \u003ccode\u003estandalone\u003c/code\u003e class, which produces a single figure without page margins or headers.\nThe \u003ccode\u003epstricks\u003c/code\u003e option enables \u003cstrong\u003ePSTricks\u003c/strong\u003e, a LaTeX package for creating PostScript-based vector graphics.\u003c/h2\u003e\u003ch3 id=\"defining-the-grid-style\" class=\"headerLink\"\u003e\n \u003ca href=\"#defining-the-grid-style\" class=\"header-mark\" aria-label=\"Direct link to section: Defining the Grid Style\"\u003e\u003c/a\u003e\u003cstrong\u003eDefining the Grid Style\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpsstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egridstyle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e gridlabels=0,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e gridwidth=6pt,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e subgriddiv=1,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e gridcolor=gray,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eDefines a new drawing style named \u003ccode\u003egridstyle\u003c/code\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003egridlabels=0\u003c/code\u003e disables coordinate labels.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003egridwidth=6pt\u003c/code\u003e sets grid line thickness.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003egridcolor=gray\u003c/code\u003e colors the grid lines gray.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003esubgriddiv=1\u003c/code\u003e ensures no subgrid divisions are drawn.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"pstricks-drawing-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#pstricks-drawing-environment\" class=\"header-mark\" aria-label=\"Direct link to section: PSTricks Drawing Environment\"\u003e\u003c/a\u003e\u003cstrong\u003ePSTricks Drawing Environment\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epspicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e(8,8)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"creates-a-drawing-area-of-8x8-units-where-all-graphical-elements-are-placed\" class=\"headerLink\"\u003e\n \u003ca href=\"#creates-a-drawing-area-of-8x8-units-where-all-graphical-elements-are-placed\" class=\"header-mark\" aria-label=\"Direct link to section: Creates a drawing area of 8x8 units where all graphical elements are placed.\"\u003e\u003c/a\u003eCreates a drawing area of 8x8 units where all graphical elements are placed.\u003c/h2\u003e\u003ch3 id=\"background-and-grid\" class=\"headerLink\"\u003e\n \u003ca href=\"#background-and-grid\" class=\"header-mark\" aria-label=\"Direct link to section: Background and Grid\"\u003e\u003c/a\u003e\u003cstrong\u003eBackground and Grid\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\psframe*\u003c/span\u003e(8,8)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\psgrid\u003c/span\u003e\u003cspan class=\"na\"\u003e[style=gridstyle]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\psframe*(8,8)\u003c/code\u003e draws a solid black square background.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\psgrid[style=gridstyle]\u003c/code\u003e overlays a gray grid on top of the background using the defined style.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"setting-line-color\" class=\"headerLink\"\u003e\n \u003ca href=\"#setting-line-color\" class=\"header-mark\" aria-label=\"Direct link to section: Setting Line Color\"\u003e\u003c/a\u003e\u003cstrong\u003eSetting Line Color\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\psset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003elinecolor=white\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"sets-the-current-drawing-color-to-white-so-that-the-subsequent-dots-will-contrast-with-the-dark-background\" class=\"headerLink\"\u003e\n \u003ca href=\"#sets-the-current-drawing-color-to-white-so-that-the-subsequent-dots-will-contrast-with-the-dark-background\" class=\"header-mark\" aria-label=\"Direct link to section: Sets the current drawing color to white, so that the subsequent dots will contrast with the dark background.\"\u003e\u003c/a\u003eSets the current drawing color to white, so that the subsequent dots will contrast with the dark background.\u003c/h2\u003e\u003ch3 id=\"drawing-white-dots-using\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-white-dots-using\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing White Dots Using multips\"\u003e\u003c/a\u003e\u003cstrong\u003eDrawing White Dots Using \u003ccode\u003emultips\u003c/code\u003e\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\multips\u003c/span\u003e(0,1)(0,1)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e7\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\multips\u003c/span\u003e(1,0)(1,0)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e7\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\qdisk\u003c/span\u003e(0,0)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e4.242pt\u003cspan class=\"nb\"\u003e}}}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eDraws the pattern of white dots forming the grid illusion:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThe first \u003ccode\u003e\\multips\u003c/code\u003e repeats the inner pattern vertically 7 times.\u003c/li\u003e\n\u003cli\u003eThe inner \u003ccode\u003e\\multips\u003c/code\u003e repeats horizontally 7 times per row.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\qdisk(0,0){4.242pt}\u003c/code\u003e draws a small white circular dot of radius ≈ 4.24pt at each grid intersection.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"closing-the-environments\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing-the-environments\" class=\"header-mark\" aria-label=\"Direct link to section: Closing the Environments\"\u003e\u003c/a\u003e\u003cstrong\u003eClosing the Environments\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epspicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"ends-both-the-drawing-and-the-document-environments-generating-the-final-figure\" class=\"headerLink\"\u003e\n \u003ca href=\"#ends-both-the-drawing-and-the-document-environments-generating-the-final-figure\" class=\"header-mark\" aria-label=\"Direct link to section: Ends both the drawing and the document environments, generating the final figure.\"\u003e\u003c/a\u003eEnds both the drawing and the document environments, generating the final figure.\u003c/h2\u003e\u003ch2 id=\"visual-and-perceptual-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-and-perceptual-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual and Perceptual Analysis\"\u003e\u003c/a\u003eVisual and Perceptual Analysis\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eObservation\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eNeural Explanation\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eDots vanish at intersections\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLateral inhibition reduces local contrast\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eDots reappear when stared at directly\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFocused vision minimizes surround inhibition\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-illusion-arises-from-lateral-inhibition--when-bright-and-dark-areas-interact-neurons-responsible-for-detecting-light-suppress-their-neighbors-activity-leading-to-false-contrast-enhancement\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-illusion-arises-from-lateral-inhibition--when-bright-and-dark-areas-interact-neurons-responsible-for-detecting-light-suppress-their-neighbors-activity-leading-to-false-contrast-enhancement\" class=\"header-mark\" aria-label=\"Direct link to section: The final output produces a dark grid where white dots seem to flicker or disappear at intersections. This illusion arises from lateral inhibition — when bright and dark areas interact, neurons responsible for detecting light suppress their neighbors’ activity, leading to false contrast enhancement.\"\u003e\u003c/a\u003eThe final output produces a dark grid where \u003cstrong\u003ewhite dots seem to flicker or disappear\u003c/strong\u003e at intersections.\nThis illusion arises from \u003cstrong\u003elateral inhibition\u003c/strong\u003e — when bright and dark areas interact, neurons responsible for detecting light suppress their neighbors’ activity, leading to \u003cstrong\u003efalse contrast enhancement\u003c/strong\u003e.\u003c/h2\u003e\u003ch2 id=\"experimental-variations\" class=\"headerLink\"\u003e\n \u003ca href=\"#experimental-variations\" class=\"header-mark\" aria-label=\"Direct link to section: Experimental Variations\"\u003e\u003c/a\u003eExperimental Variations\u003c/h2\u003e\u003cp\u003eYou can adjust the parameters in the code to explore perceptual effects:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eParameter\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003ePurpose\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Effect\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003egridwidth\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eGrid line thickness\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAlters strength of illusion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003egridcolor\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eGrid color\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAffects contrast intensity\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eDot size (\u003ccode\u003e\\qdisk\u003c/code\u003e)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eControls dot visibility\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLarger dots reduce illusion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eNumber of repetitions\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChanges grid density\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMore intersections strengthen illusion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eExample modification:\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\psgrid\u003c/span\u003e\u003cspan class=\"na\"\u003e[style=gridstyle,gridcolor=lightgray]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\qdisk\u003c/span\u003e(0,0)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"scientific-and-artistic-relevance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scientific-and-artistic-relevance\" class=\"header-mark\" aria-label=\"Direct link to section: Scientific and Artistic Relevance\"\u003e\u003c/a\u003eScientific and Artistic Relevance\u003c/h2\u003e\u003cp\u003eThe Hermann Grid Illusion bridges \u003cstrong\u003evisual neuroscience\u003c/strong\u003e, \u003cstrong\u003emathematics\u003c/strong\u003e, and \u003cstrong\u003eart\u003c/strong\u003e.\nIt demonstrates how simple geometry can expose deep insights into \u003cstrong\u003ehuman perception\u003c/strong\u003e. Applications include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eCognitive psychology\u003c/strong\u003e — studying spatial inhibition and lightness perception.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eVisual design\u003c/strong\u003e — understanding how contrast guides attention.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMathematical art\u003c/strong\u003e — turning perceptual science into geometric beauty.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"by-using-latex-pstricks-we-can-mathematically-and-aesthetically-recreate-this-effect-merging-science-and-art-in-a-single-figure\" class=\"headerLink\"\u003e\n \u003ca href=\"#by-using-latex-pstricks-we-can-mathematically-and-aesthetically-recreate-this-effect-merging-science-and-art-in-a-single-figure\" class=\"header-mark\" aria-label=\"Direct link to section: The White Dot Illusion exemplifies how perception is context-dependent, not purely physical. By using LaTeX PSTricks, we can mathematically and aesthetically recreate this effect, merging science and art in a single figure.\"\u003e\u003c/a\u003eThe White Dot Illusion exemplifies how perception is \u003cstrong\u003econtext-dependent\u003c/strong\u003e, not purely physical.\nBy using \u003cstrong\u003eLaTeX PSTricks\u003c/strong\u003e, we can mathematically and aesthetically recreate this effect, merging \u003cstrong\u003escience and art\u003c/strong\u003e in a single figure.\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eHermann, L. (1870). \u003cem\u003eEine Erscheinung simultanen Contrastes.\u003c/em\u003e Pflügers Archiv für die gesamte Physiologie.\u003c/li\u003e\n\u003cli\u003eCoren, S., \u0026amp; Girgus, J. S. (1978). \u003cem\u003eSeeing is Deceiving: The Psychology of Visual Illusions.\u003c/em\u003e Lawrence Erlbaum.\u003c/li\u003e\n\u003cli\u003eGregory, R. L. (1997). \u003cem\u003eEye and Brain: The Psychology of Seeing.\u003c/em\u003e Oxford University Press.\u003c/li\u003e\n\u003cli\u003ePSTricks User Manual, Version 3.1.8 (2023).\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "The Circle Illusion: The Perceptual Phenomenon of Size Around a Central Circle", "date_published" : "2025-10-31T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/circle-illusion/", "url" : "https://www.aantriono.com/en/circle-illusion/", "summary": "This article discusses the circle illusion, where two identical central circles appear different in size due to surrounding visual context. Built with LaTeX TikZ, this article provides a complete explanation of the code structure and perceptual science behind it.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the circle illusion, where two identical central circles appear different in size due to surrounding visual context. Built with LaTeX TikZ, this article provides a complete explanation of the code structure and perceptual science behind it.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"this-article-demonstrates-the-phenomenon-using-latex-tikz-providing-a-detailed-explanation-of-each-code-component-and-the-scientific-concept-behind-it\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-article-demonstrates-the-phenomenon-using-latex-tikz-providing-a-detailed-explanation-of-each-code-component-and-the-scientific-concept-behind-it\" class=\"header-mark\" aria-label=\"Direct link to section: In the realm of visual perception, the human brain often processes size relatively rather than absolutely. One of the most well-known examples of this is the Ebbinghaus Illusion, or Circle Illusion, in which two identical circles appear to be of different sizes due to the surrounding shapes. This article demonstrates the phenomenon using LaTeX TikZ, providing a detailed explanation of each code component and the scientific concept behind it.\"\u003e\u003c/a\u003eIn the realm of visual perception, the human brain often processes size \u003cstrong\u003erelatively\u003c/strong\u003e rather than \u003cstrong\u003eabsolutely\u003c/strong\u003e.\nOne of the most well-known examples of this is the \u003cstrong\u003eEbbinghaus Illusion\u003c/strong\u003e, or \u003cstrong\u003eCircle Illusion\u003c/strong\u003e, in which two identical circles appear to be of different sizes due to the surrounding shapes.\nThis article demonstrates the phenomenon using \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, providing a detailed explanation of each code component and the scientific concept behind it.\u003c/h2\u003e\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eBelow is the complete LaTeX code to generate two orange central circles that appear different in size due to variations in the surrounding blue circles.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) 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48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[baseline=(X.base)]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[circle,fill=orange,draw=orange,minimum size=2cm]\u003c/span\u003e (X) at (0,0) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,60,...,330\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!50!white]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e:3.4) circle (1.6);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[baseline=(X.base)]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[circle,fill=orange,draw=orange,minimum size=2cm]\u003c/span\u003e (X) at (0,0) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,45,...,360\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!50!white]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e:1.5) circle (.4);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/circle-illusion/lingkaran.webp\" title=\"ilusi\" data-thumbnail=\"/circle-illusion/lingkaran.webp\" data-sub-html=\"\u003ch2\u003e Orange circles are the same size or different \u003c/h2\u003e\u003cp\u003eilusi\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/circle-illusion/lingkaran.webp' srcset=\"/circle-illusion/lingkaran_hu_f0c1db922a9fd1d9.webp 800w, /circle-illusion/lingkaran_hu_3806f182842386ab.webp 1200w, /circle-illusion/lingkaran_hu_554c179c03f8354d.webp 1600w\" alt=\"ilusi\" height=\"391\" width=\"618\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003e Orange circles are the same size or different \u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch2 id=\"explanation-of-the-code-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-of-the-code-structure\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation of the Code Structure\"\u003e\u003c/a\u003eExplanation of the Code Structure\u003c/h2\u003e\u003ch3 id=\"document-declaration\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-declaration\" class=\"header-mark\" aria-label=\"Direct link to section: Document Declaration\"\u003e\u003c/a\u003e\u003cstrong\u003eDocument Declaration\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"its-ideal-for-generating-standalone-figures-or-mathematical-graphics\" class=\"headerLink\"\u003e\n \u003ca href=\"#its-ideal-for-generating-standalone-figures-or-mathematical-graphics\" class=\"header-mark\" aria-label=\"Direct link to section: The standalone document class ensures that only the drawing is rendered—no page margins, headers, or text. It’s ideal for generating standalone figures or mathematical graphics.\"\u003e\u003c/a\u003eThe \u003ccode\u003estandalone\u003c/code\u003e document class ensures that only the drawing is rendered—no page margins, headers, or text.\nIt’s ideal for generating standalone figures or mathematical graphics.\u003c/h2\u003e\u003ch3 id=\"importing-the-tikz-package\" class=\"headerLink\"\u003e\n \u003ca href=\"#importing-the-tikz-package\" class=\"header-mark\" aria-label=\"Direct link to section: Importing the TikZ Package\"\u003e\u003c/a\u003e\u003cstrong\u003eImporting the TikZ Package\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"marks-the-beginning-of-the-document-body-where-the-drawings-and-commands-are-placed\" class=\"headerLink\"\u003e\n \u003ca href=\"#marks-the-beginning-of-the-document-body-where-the-drawings-and-commands-are-placed\" class=\"header-mark\" aria-label=\"Direct link to section: Marks the beginning of the document body where the drawings and commands are placed.\"\u003e\u003c/a\u003eMarks the beginning of the document body where the drawings and commands are placed.\u003c/h2\u003e\u003ch3 id=\"first-illustration--left-circle-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#first-illustration--left-circle-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: First Illustration — Left Circle Illusion\"\u003e\u003c/a\u003e\u003cstrong\u003eFirst Illustration — Left Circle Illusion\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[baseline=(X.base)]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"the-baselinexbase-ensures-that-both-drawings-align-vertically-when-placed-side-by-side\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-baselinexbase-ensures-that-both-drawings-align-vertically-when-placed-side-by-side\" class=\"header-mark\" aria-label=\"Direct link to section: This environment defines a TikZ drawing area. The baseline=(X.base) ensures that both drawings align vertically when placed side by side.\"\u003e\u003c/a\u003eThis environment defines a TikZ drawing area.\nThe \u003ccode\u003ebaseline=(X.base)\u003c/code\u003e ensures that both drawings align vertically when placed side by side.\u003c/h2\u003e\u003ch4 id=\"central-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#central-circle\" class=\"header-mark\" aria-label=\"Direct link to section: Central Circle\"\u003e\u003c/a\u003e\u003cstrong\u003eCentral Circle\u003c/strong\u003e\u003c/h4\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[circle,fill=orange,draw=orange,minimum size=2cm]\u003c/span\u003e (X) at (0,0) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eCreates a \u003cstrong\u003e2 cm diameter orange circle\u003c/strong\u003e at the coordinate (0,0).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e(X)\u003c/code\u003e labels the node for reference.\u003c/li\u003e\n\u003cli\u003eActs as the \u003cstrong\u003ecentral reference circle\u003c/strong\u003e for the illusion.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch4 id=\"surrounding-large-circles\" class=\"headerLink\"\u003e\n \u003ca href=\"#surrounding-large-circles\" class=\"header-mark\" aria-label=\"Direct link to section: Surrounding Large Circles\"\u003e\u003c/a\u003e\u003cstrong\u003eSurrounding Large Circles\u003c/strong\u003e\u003c/h4\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,60,...,330\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!50!white]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e:3.4) circle (1.6);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eDraws \u003cstrong\u003esix large blue circles\u003c/strong\u003e spaced 60° apart.\u003c/li\u003e\n\u003cli\u003eEach is 3.4 cm away from the center and has a radius of 1.6 cm.\u003c/li\u003e\n\u003cli\u003eColor: a 50% mix of blue and white for visual softness.\nResult: The orange center appears \u003cstrong\u003esmaller\u003c/strong\u003e due to large outer circles.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"horizontal-spacing-between-figures\" class=\"headerLink\"\u003e\n \u003ca href=\"#horizontal-spacing-between-figures\" class=\"header-mark\" aria-label=\"Direct link to section: Horizontal Spacing Between Figures\"\u003e\u003c/a\u003e\u003cstrong\u003eHorizontal Spacing Between Figures\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"adds-1-cm-of-horizontal-space-between-the-two-figures-so-they-dont-overlap\" class=\"headerLink\"\u003e\n \u003ca href=\"#adds-1-cm-of-horizontal-space-between-the-two-figures-so-they-dont-overlap\" class=\"header-mark\" aria-label=\"Direct link to section: Adds 1 cm of horizontal space between the two figures so they don’t overlap.\"\u003e\u003c/a\u003eAdds 1 cm of horizontal space between the two figures so they don’t overlap.\u003c/h2\u003e\u003ch3 id=\"second-illustration--right-circle-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#second-illustration--right-circle-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: Second Illustration — Right Circle Illusion\"\u003e\u003c/a\u003e\u003cstrong\u003eSecond Illustration — Right Circle Illusion\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[baseline=(X.base)]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"begins-the-second-tikz-environment-for-the-right-hand-side-illustration\" class=\"headerLink\"\u003e\n \u003ca href=\"#begins-the-second-tikz-environment-for-the-right-hand-side-illustration\" class=\"header-mark\" aria-label=\"Direct link to section: Begins the second TikZ environment for the right-hand side illustration.\"\u003e\u003c/a\u003eBegins the second TikZ environment for the right-hand side illustration.\u003c/h2\u003e\u003ch4 id=\"central-circle-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#central-circle-1\" class=\"header-mark\" aria-label=\"Direct link to section: Central Circle\"\u003e\u003c/a\u003e\u003cstrong\u003eCentral Circle\u003c/strong\u003e\u003c/h4\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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Despite being the same size, it will appear larger in this visual context.\"\u003e\u003c/a\u003eIdentical to the first central circle. Despite being the same size, it will appear \u003cstrong\u003elarger\u003c/strong\u003e in this visual context.\u003c/h2\u003e\u003ch4 id=\"surrounding-small-circles\" class=\"headerLink\"\u003e\n \u003ca href=\"#surrounding-small-circles\" class=\"header-mark\" aria-label=\"Direct link to section: Surrounding Small Circles\"\u003e\u003c/a\u003e\u003cstrong\u003eSurrounding Small Circles\u003c/strong\u003e\u003c/h4\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,45,...,360\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue!50!white]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e:1.5) circle (.4);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eDraws \u003cstrong\u003eeight small blue circles\u003c/strong\u003e, each spaced 45° apart.\u003c/li\u003e\n\u003cli\u003eEach is 1.5 cm from the center and has a radius of only 0.4 cm.\nResult: The central orange circle now appears \u003cstrong\u003elarger\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"end-of-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#end-of-document\" class=\"header-mark\" aria-label=\"Direct link to section: End of Document\"\u003e\u003c/a\u003e\u003cstrong\u003eEnd of Document\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"marks-the-end-of-the-latex-document-the-output-will-show-two-side-by-side-illustrations-forming-the-ebbinghaus-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#marks-the-end-of-the-latex-document-the-output-will-show-two-side-by-side-illustrations-forming-the-ebbinghaus-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: Marks the end of the LaTeX document. The output will show two side-by-side illustrations forming the Ebbinghaus illusion.\"\u003e\u003c/a\u003eMarks the end of the LaTeX document. The output will show two side-by-side illustrations forming the Ebbinghaus illusion.\u003c/h2\u003e\u003ch2 id=\"visual-analysis\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-analysis\" class=\"header-mark\" aria-label=\"Direct link to section: Visual Analysis\"\u003e\u003c/a\u003eVisual Analysis\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: left\"\u003eFigure\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003eSurrounding Circle Size\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003eDistance from Center\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003ePerceived Effect\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eLeft\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eLarge (1.6 cm radius)\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003e3.4 cm\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eCentral circle looks smaller\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eRight\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eSmall (0.4 cm radius)\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003e1.5 cm\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eCentral circle looks larger\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003eEven though both orange circles are identical, the brain \u003cstrong\u003einterprets size contextually\u003c/strong\u003e, not absolutely.\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"how-the-illusion-works\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-the-illusion-works\" class=\"header-mark\" aria-label=\"Direct link to section: How the Illusion Works\"\u003e\u003c/a\u003eHow the Illusion Works\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eContextual Comparison\u003c/strong\u003e — The brain interprets object size relative to surrounding objects.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eVisual Normalization\u003c/strong\u003e — Perceptual systems normalize differences to simplify spatial processing.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eEnvironmental Cues\u003c/strong\u003e — Large surroundings shrink perception; small surroundings enlarge it.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"further-experiments\" class=\"headerLink\"\u003e\n \u003ca href=\"#further-experiments\" class=\"header-mark\" aria-label=\"Direct link to section: Further Experiments\"\u003e\u003c/a\u003eFurther Experiments\u003c/h2\u003e\u003cp\u003eModify the following parameters in the TikZ code to explore different effects:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eParameter\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003ePurpose\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Effect\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e3.4\u003c/code\u003e / \u003ccode\u003e1.5\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eDistance from center\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChanges overall spacing\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e1.6\u003c/code\u003e / \u003ccode\u003e0.4\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eSurrounding circle size\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eStrengthens or weakens illusion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eColor\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChange \u003ccode\u003eblue!50!white\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAlters color contrast and depth\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eCircle count\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChange 60° / 45° steps\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAffects density and symmetry\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eExample variation:\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,30,...,330\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[green!40!white]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e:2.5) circle (.8);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"scientific-and-artistic-relevance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scientific-and-artistic-relevance\" class=\"header-mark\" aria-label=\"Direct link to section: Scientific and Artistic Relevance\"\u003e\u003c/a\u003eScientific and Artistic Relevance\u003c/h2\u003e\u003cp\u003eThe circle illusion bridges \u003cstrong\u003epsychology, geometry, and art\u003c/strong\u003e.\nIt provides valuable insights and applications in:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eCognitive psychology\u003c/strong\u003e — understanding size perception and context.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDesign \u0026amp; UI/UX\u003c/strong\u003e — using space and contrast for visual emphasis.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMathematical art\u003c/strong\u003e — visualizing relationships between symmetry and illusion.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"using-latex-tikz-we-can-mathematically-recreate-this-visual-phenomenon-blending-science-and-aesthetics-into-a-single-elegant-representation\" class=\"headerLink\"\u003e\n \u003ca href=\"#using-latex-tikz-we-can-mathematically-recreate-this-visual-phenomenon-blending-science-and-aesthetics-into-a-single-elegant-representation\" class=\"header-mark\" aria-label=\"Direct link to section: The Ebbinghaus Illusion reveals that human perception of size is context-dependent, not objective. Using LaTeX TikZ, we can mathematically recreate this visual phenomenon, blending science and aesthetics into a single, elegant representation.\"\u003e\u003c/a\u003eThe Ebbinghaus Illusion reveals that human perception of size is \u003cstrong\u003econtext-dependent\u003c/strong\u003e, not objective.\nUsing \u003cstrong\u003eLaTeX TikZ\u003c/strong\u003e, we can mathematically recreate this visual phenomenon, blending \u003cstrong\u003escience and aesthetics\u003c/strong\u003e into a single, elegant representation.\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eEbbinghaus, H. (1897). \u003cem\u003eÜber eine neue Methode zur Prüfung geistiger Fähigkeiten und ihre Anwendung bei Schulkindern.\u003c/em\u003e\u003c/li\u003e\n\u003cli\u003eCoren, S., \u0026amp; Girgus, J. S. (1978). \u003cem\u003eSeeing is Deceiving: The Psychology of Visual Illusions.\u003c/em\u003e Lawrence Erlbaum.\u003c/li\u003e\n\u003cli\u003eGregory, R. L. (1997). \u003cem\u003eEye and Brain: The Psychology of Seeing.\u003c/em\u003e Oxford University Press.\u003c/li\u003e\n\u003cli\u003ePGF/TikZ Manual, Version 3.1.10 (2023).\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "The Circle Illusion: Between Geometry and Visual Perception", "date_published" : "2025-10-31T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/optical-illusion/", "url" : "https://www.aantriono.com/en/optical-illusion/", "summary": "This article explores how an arrangement of squares can form a rotating circular illusion using LaTeX and TikZ. We will examine the structure of the code, the visual principles behind the illusion, and its significance in perception and visual art.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explores how an \u003cstrong\u003earrangement of squares\u003c/strong\u003e can form a rotating circular illusion using \u003cstrong\u003eLaTeX and TikZ\u003c/strong\u003e. We will examine the structure of the code, the visual principles behind the illusion, and its significance in perception and visual art.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"this-article-explores-how-an-arrangement-of-squares-can-form-a-rotating-circular-illusion-using-latex-and-tikz-we-will-examine-the-structure-of-the-code-the-visual-principles-behind-the-illusion-and-its-significance-in-perception-and-visual-art\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-article-explores-how-an-arrangement-of-squares-can-form-a-rotating-circular-illusion-using-latex-and-tikz-we-will-examine-the-structure-of-the-code-the-visual-principles-behind-the-illusion-and-its-significance-in-perception-and-visual-art\" class=\"header-mark\" aria-label=\"Direct link to section: In the world of visual perception, optical illusions are fascinating phenomena that reveal how the human brain interprets visual information. One such illusion is the circle illusion, where simple geometric patterns can create the impression of motion or form that does not actually exist. This article explores how an arrangement of squares can form a rotating circular illusion using LaTeX and TikZ. We will examine the structure of the code, the visual principles behind the illusion, and its significance in perception and visual art.\"\u003e\u003c/a\u003eIn the world of visual perception, \u003cstrong\u003eoptical illusions\u003c/strong\u003e are fascinating phenomena that reveal how the human brain interprets visual information. One such illusion is the \u003cstrong\u003ecircle illusion\u003c/strong\u003e, where simple geometric patterns can create the impression of motion or form that does not actually exist.\nThis article explores how an \u003cstrong\u003earrangement of squares\u003c/strong\u003e can form a rotating circular illusion using \u003cstrong\u003eLaTeX and TikZ\u003c/strong\u003e. We will examine the structure of the code, the visual principles behind the illusion, and its significance in perception and visual art.\u003c/h2\u003e\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eThe following image is generated using \u003cstrong\u003eTikZ\u003c/strong\u003e, a powerful LaTeX graphics library designed for mathematical and scientific illustrations.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=black!40!white]\u003c/span\u003e (-6,-6) rectangle (6,6);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\twist\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e70/5/12,56/4/-12,42/3/12,28/2/-12\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\m\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,3,...,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [thick,color=white,shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,rotate=\u003cspan class=\"k\"\u003e\\twist\u003c/span\u003e+360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-.15,-.15) rectangle (.15,.15);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\m\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2,4,...,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [thick,color=black,shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,rotate=\u003cspan class=\"k\"\u003e\\twist\u003c/span\u003e+360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-.15,-.15) rectangle (.15,.15);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output produced:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/optical-illusion/ilusi.webp\" title=\"ilusi\" data-thumbnail=\"/optical-illusion/ilusi.webp\" data-sub-html=\"\u003ch2\u003e Circle or Spiral \u003c/h2\u003e\u003cp\u003eilusi\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/optical-illusion/ilusi.webp' srcset=\"/optical-illusion/ilusi_hu_2a2e2162ea3f4112.webp 800w, /optical-illusion/ilusi_hu_51b73819bb84c62f.webp 1200w, /optical-illusion/ilusi_hu_1c92fbbd77a5eb7f.webp 1600w\" alt=\"ilusi\" height=\"471\" width=\"473\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003e Circle or Spiral \u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch2 id=\"code-structure-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#code-structure-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Code Structure Explanation\"\u003e\u003c/a\u003eCode Structure Explanation\u003c/h2\u003e\u003cp\u003eThis section explains each component of the code so you can understand how the pattern is constructed step by step.\u003c/p\u003e\n\u003ch3 id=\"document-class\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-class\" class=\"header-mark\" aria-label=\"Direct link to section: Document Class\"\u003e\u003c/a\u003e\u003cstrong\u003eDocument Class\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"the-standalone-class-allows-you-to-compile-only-the-figure-without-page-margins-or-text-this-is-ideal-for-creating-diagrams-or-standalone-graphics\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-standalone-class-allows-you-to-compile-only-the-figure-without-page-margins-or-text-this-is-ideal-for-creating-diagrams-or-standalone-graphics\" class=\"header-mark\" aria-label=\"Direct link to section: The standalone class allows you to compile only the figure without page margins or text. This is ideal for creating diagrams or standalone graphics.\"\u003e\u003c/a\u003eThe \u003ccode\u003estandalone\u003c/code\u003e class allows you to compile only the figure without page margins or text. This is ideal for creating diagrams or standalone graphics.\u003c/h2\u003e\u003ch3 id=\"loading-the-tikz-package\" class=\"headerLink\"\u003e\n \u003ca href=\"#loading-the-tikz-package\" class=\"header-mark\" aria-label=\"Direct link to section: Loading the TikZ Package\"\u003e\u003c/a\u003e\u003cstrong\u003eLoading the TikZ Package\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"tikz-short-for--is-a-latex-library-for-vector-graphics-providing-precise-control-over-shapes-colors-and-geometric-transformations\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-short-for--is-a-latex-library-for-vector-graphics-providing-precise-control-over-shapes-colors-and-geometric-transformations\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ (short for \u0026ldquo;TikZ ist kein Zeichenprogramm\u0026rdquo;) is a LaTeX library for vector graphics, providing precise control over shapes, colors, and geometric transformations.\"\u003e\u003c/a\u003eTikZ (short for \u003cem\u003e\u0026ldquo;TikZ ist kein Zeichenprogramm\u0026rdquo;\u003c/em\u003e) is a LaTeX library for vector graphics, providing precise control over shapes, colors, and geometric transformations.\u003c/h2\u003e\u003ch3 id=\"tikz-picture-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-picture-environment\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Picture Environment\"\u003e\u003c/a\u003e\u003cstrong\u003eTikZ Picture Environment\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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Within this block, we define all visual elements—background, shapes, colors, and loops.\"\u003e\u003c/a\u003eAll TikZ drawing commands are placed inside this environment. Within this block, we define all visual elements—background, shapes, colors, and loops.\u003c/h2\u003e\u003ch3 id=\"background\" class=\"headerLink\"\u003e\n \u003ca href=\"#background\" class=\"header-mark\" aria-label=\"Direct link to section: Background\"\u003e\u003c/a\u003e\u003cstrong\u003eBackground\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[color=black!40!white]\u003c/span\u003e (-6,-6) rectangle (6,6);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-gray-background-enhances-contrast-between-black-and-white-squares\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-gray-background-enhances-contrast-between-black-and-white-squares\" class=\"header-mark\" aria-label=\"Direct link to section: The \\fill command fills a rectangular area with a color. black!40!white means 40% black mixed with white, producing a neutral gray background. Coordinates (-6,-6) to (6,6) define the drawing area. This gray background enhances contrast between black and white squares.\"\u003e\u003c/a\u003eThe \u003ccode\u003e\\fill\u003c/code\u003e command fills a rectangular area with a color.\n\u003ccode\u003eblack!40!white\u003c/code\u003e means \u003cstrong\u003e40% black mixed with white\u003c/strong\u003e, producing a neutral gray background.\nCoordinates \u003ccode\u003e(-6,-6)\u003c/code\u003e to \u003ccode\u003e(6,6)\u003c/code\u003e define the drawing area.\nThis gray background enhances contrast between black and white squares.\u003c/h2\u003e\u003ch3 id=\"main-loop-for-circular-layers\" class=\"headerLink\"\u003e\n \u003ca href=\"#main-loop-for-circular-layers\" class=\"header-mark\" aria-label=\"Direct link to section: Main Loop for Circular Layers\"\u003e\u003c/a\u003e\u003cstrong\u003eMain Loop for Circular Layers\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\twist\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e70/5/12,56/4/-12,42/3/12,28/2/-12\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis line defines the core structure of the pattern.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\n\u003c/code\u003e = number of squares in one ring\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\r\u003c/code\u003e = radius (distance from the center)\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\twist\u003c/code\u003e = rotation angle for each layer\nWith four combinations, we get \u003cstrong\u003efour concentric circular layers\u003c/strong\u003e, each with a slightly different rotation. The alternating \u003ccode\u003e12\u003c/code\u003e and \u003ccode\u003e-12\u003c/code\u003e twist values make the outer and inner layers appear to rotate in opposite directions.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"white-squares\" class=\"headerLink\"\u003e\n \u003ca href=\"#white-squares\" class=\"header-mark\" aria-label=\"Direct link to section: White Squares\"\u003e\u003c/a\u003e\u003cstrong\u003eWhite Squares\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\m\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,3,...,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [thick,color=white,shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,rotate=\u003cspan class=\"k\"\u003e\\twist\u003c/span\u003e+360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-.15,-.15) rectangle (.15,.15);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis loop draws \u003cstrong\u003ewhite squares\u003c/strong\u003e at odd positions (1, 3, 5, \u0026hellip;).\nExplanation:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e360/\\n*\\m\u003c/code\u003e determines each square’s angular position.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eshift={(...:\\r)}\u003c/code\u003e places the square along the circle’s radius.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003erotate=\\twist+360/\\n*\\m\u003c/code\u003e sets the orientation of each square.\u003c/li\u003e\n\u003cli\u003eThe coordinates define the square’s size centered at its position.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch3 id=\"black-squares\" class=\"headerLink\"\u003e\n \u003ca href=\"#black-squares\" class=\"header-mark\" aria-label=\"Direct link to section: Black Squares\"\u003e\u003c/a\u003e\u003cstrong\u003eBlack Squares\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\m\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2,4,...,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [thick,color=black,shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,rotate=\u003cspan class=\"k\"\u003e\\twist\u003c/span\u003e+360/\u003cspan class=\"k\"\u003e\\n*\\m\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (-.15,-.15) rectangle (.15,.15);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-loop-draws-black-squares-in-even-positions-alternating-with-the-white-ones-the-contrast-between-them-is-the-foundation-of-the-optical-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-loop-draws-black-squares-in-even-positions-alternating-with-the-white-ones-the-contrast-between-them-is-the-foundation-of-the-optical-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: This loop draws black squares in even positions, alternating with the white ones. The contrast between them is the foundation of the optical illusion.\"\u003e\u003c/a\u003eThis loop draws \u003cstrong\u003eblack squares\u003c/strong\u003e in even positions, alternating with the white ones. The contrast between them is the foundation of the optical illusion.\u003c/h2\u003e\u003ch3 id=\"closing-the-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing-the-environment\" class=\"header-mark\" aria-label=\"Direct link to section: Closing the Environment\"\u003e\u003c/a\u003e\u003cstrong\u003eClosing the Environment\u003c/strong\u003e\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"these-commands-close-the-tikz-and-latex-document-environments-the-result-is-a-symmetric-composition-of-black-and-white-square-rings-that-appear-to-rotate-when-viewed-carefully\" class=\"headerLink\"\u003e\n \u003ca href=\"#these-commands-close-the-tikz-and-latex-document-environments-the-result-is-a-symmetric-composition-of-black-and-white-square-rings-that-appear-to-rotate-when-viewed-carefully\" class=\"header-mark\" aria-label=\"Direct link to section: These commands close the TikZ and LaTeX document environments. The result is a symmetric composition of black and white square rings that appear to rotate when viewed carefully.\"\u003e\u003c/a\u003eThese commands close the TikZ and LaTeX document environments. The result is \u003cstrong\u003ea symmetric composition of black and white square rings\u003c/strong\u003e that appear to rotate when viewed carefully.\u003c/h2\u003e\u003ch2 id=\"the-optical-illusion-explained\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-optical-illusion-explained\" class=\"header-mark\" aria-label=\"Direct link to section: The Optical Illusion Explained\"\u003e\u003c/a\u003eThe Optical Illusion Explained\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eColor Contrast\u003c/strong\u003e — The alternating black and white squares cause high luminance contrast, triggering visual sensitivity in the retina.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eGradual Rotation\u003c/strong\u003e — The slightly different twist angles between layers create the illusion of counter-rotating rings.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMicrosaccades\u003c/strong\u003e — Tiny involuntary eye movements make static patterns appear to move, reinforcing the illusion.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"experimenting-with-parameters\" class=\"headerLink\"\u003e\n \u003ca href=\"#experimenting-with-parameters\" class=\"header-mark\" aria-label=\"Direct link to section: Experimenting with Parameters\"\u003e\u003c/a\u003eExperimenting with Parameters\u003c/h2\u003e\u003cp\u003eYou can modify several parameters to create new visual effects:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eParameter\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eDescription\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eVisual Impact\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\r\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCircle radius\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eChanges ring size\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\n\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNumber of squares\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAlters pattern density\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\twist\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eRotation angle\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdjusts illusion direction and strength\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eColor\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eOther color pairs\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAffects depth and vibrancy\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eExample modification:\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\r\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\twist\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e60/5/10,50/4/-10,40/3/10,30/2/-10\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-creates-a-softer-illusion-with-reduced-contrast\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-creates-a-softer-illusion-with-reduced-contrast\" class=\"header-mark\" aria-label=\"Direct link to section: This creates a softer illusion with reduced contrast.\"\u003e\u003c/a\u003eThis creates a softer illusion with reduced contrast.\u003c/h2\u003e\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"with-tikz-latex-becomes-not-only-a-scientific-writing-tool-but-also-a-creative-medium-bridging-art-geometry-and-psychology\" class=\"headerLink\"\u003e\n \u003ca href=\"#with-tikz-latex-becomes-not-only-a-scientific-writing-tool-but-also-a-creative-medium-bridging-art-geometry-and-psychology\" class=\"header-mark\" aria-label=\"Direct link to section: The circle illusion beautifully demonstrates how simple geometry and visual contrast can manipulate human perception. With TikZ, LaTeX becomes not only a scientific writing tool but also a creative medium bridging art, geometry, and psychology.\"\u003e\u003c/a\u003eThe circle illusion beautifully demonstrates how \u003cstrong\u003esimple geometry\u003c/strong\u003e and \u003cstrong\u003evisual contrast\u003c/strong\u003e can manipulate human perception.\nWith TikZ, LaTeX becomes not only a scientific writing tool but also a \u003cstrong\u003ecreative medium\u003c/strong\u003e bridging art, geometry, and psychology.\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eGregory, R. L. (1997). \u003cem\u003eEye and Brain: The Psychology of Seeing\u003c/em\u003e. Oxford University Press.\u003c/li\u003e\n\u003cli\u003eKitaoka, A. (2003). \u003cem\u003eRotating Snakes Illusion\u003c/em\u003e. \u003ca href=\"https://www.ritsumei.ac.jp/~akitaoka/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAkiyoshi Kitaoka’s Optical Illusions\u003c/a\u003e.\u003c/li\u003e\n\u003cli\u003eWare, C. (2013). \u003cem\u003eInformation Visualization: Perception for Design\u003c/em\u003e. Morgan Kaufmann.\u003c/li\u003e\n\u003cli\u003eThe PGF/TikZ Manual, Version 3.1.10 (2023).\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cstrong\u003eWritten by:\u003c/strong\u003e \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAan Triono\u003c/a\u003e\n\u003cstrong\u003eLicense:\u003c/strong\u003e CC BY-SA 4.0\u003c/p\u003e", "language": "en" }, { "title" : "Technical Guide: How to Compile LaTeX Documents Completely and Efficiently", "date_published" : "2025-10-17T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/compile-document-in-latex/", "url" : "https://www.aantriono.com/en/compile-document-in-latex/", "summary": "Compilation is the process of translating LaTeX commands into a ready-to-print document. This article discusses how LaTeX compilation works, types of compilers available, the compilation sequence, and how to resolve errors efficiently.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eCompilation is the process of translating LaTeX commands into a ready-to-print document.\nThis article discusses how LaTeX compilation works, types of compilers available, the compilation sequence, and how to resolve errors efficiently.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"hahahugoshortcode117s0hbhb\" class=\"headerLink\"\u003e\n \u003ca href=\"#hahahugoshortcode117s0hbhb\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX is a document preparation system that produces high-quality output, especially for scientific and technical text. Unlike conventional word processors, a LaTeX document must be compiled before a .tex file can be converted into the final format such as .pdf, .dvi, or .ps. \u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cstrong\u003eNote:\u003c/strong\u003e This article was generated with the help of AI technology and has been manually reviewed to ensure accuracy and clarity.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\"\u003e\u003c/a\u003eLaTeX is a document preparation system that produces high-quality output, especially for scientific and technical text.\nUnlike conventional word processors, a LaTeX document \u003cstrong\u003emust be compiled\u003c/strong\u003e before a \u003ccode\u003e.tex\u003c/code\u003e file can be converted into the final format such as \u003ccode\u003e.pdf\u003c/code\u003e, \u003ccode\u003e.dvi\u003c/code\u003e, or \u003ccode\u003e.ps\u003c/code\u003e.\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cstrong\u003eNote:\u003c/strong\u003e This article was generated with the help of AI technology and has been manually reviewed to ensure accuracy and clarity.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/h2\u003e\u003ch2 id=\"what-is-latex-compilation\" class=\"headerLink\"\u003e\n \u003ca href=\"#what-is-latex-compilation\" class=\"header-mark\" aria-label=\"Direct link to section: What is LaTeX Compilation?\"\u003e\u003c/a\u003eWhat is LaTeX Compilation?\u003c/h2\u003e\u003cp\u003eLaTeX compilation is the process performed by a \u003cem\u003ecompiler\u003c/em\u003e to read the source file (usually with the \u003ccode\u003e.tex\u003c/code\u003e extension), interpret the commands inside (\u003ccode\u003e\\section{}\u003c/code\u003e, \u003ccode\u003e\\usepackage{}\u003c/code\u003e, etc.), and produce an output file such as \u003ccode\u003e.pdf\u003c/code\u003e.\nIn short:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003edocument.tex → [compiler] → document.pdf\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-process-differs-from-a-typical-word-processor-like-microsoft-word-because-latex-separates-content-writing-from-output-rendering\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-process-differs-from-a-typical-word-processor-like-microsoft-word-because-latex-separates-content-writing-from-output-rendering\" class=\"header-mark\" aria-label=\"Direct link to section: This process differs from a typical word processor like Microsoft Word because LaTeX separates content writing from output rendering.\"\u003e\u003c/a\u003eThis process differs from a typical \u003cem\u003eword processor\u003c/em\u003e like Microsoft Word because LaTeX separates \u003cstrong\u003econtent writing\u003c/strong\u003e from \u003cstrong\u003eoutput rendering\u003c/strong\u003e.\u003c/h2\u003e\u003ch2 id=\"types-of-latex-compilers\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-latex-compilers\" class=\"header-mark\" aria-label=\"Direct link to section: Types of LaTeX Compilers\"\u003e\u003c/a\u003eTypes of LaTeX Compilers\u003c/h2\u003e\u003cp\u003eHere are some commonly used LaTeX compilers:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eCompiler\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eMain Output\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eAdvantages\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eDisadvantages\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003epdfLaTeX\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ePDF directly\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFast, stable, compatible with most packages\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLimited to TeX fonts and ASCII input\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eXeLaTeX\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ePDF\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eSupports Unicode and system fonts (TTF/OTF)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eSlightly slower\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eLuaLaTeX\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ePDF\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eSupports Lua scripting, high performance\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLess stable with older packages\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eLaTeX → DVI → PS → PDF\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ePDF/DVI\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eClassic method, suitable for legacy projects\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLong process and rarely used today\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cblockquote\u003e\n \u003cp\u003e💡 \u003cstrong\u003eRecommendation:\u003c/strong\u003e Use \u003ccode\u003epdfLaTeX\u003c/code\u003e for standard documents, \u003ccode\u003eXeLaTeX\u003c/code\u003e for non-Latin languages (e.g., Indonesian, Japanese, Arabic), and \u003ccode\u003eLuaLaTeX\u003c/code\u003e for projects involving advanced graphics or scripting.\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"the-compilation-process\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-compilation-process\" class=\"header-mark\" aria-label=\"Direct link to section: The Compilation Process\"\u003e\u003c/a\u003eThe Compilation Process\u003c/h2\u003e\u003cp\u003eWhen compiling, LaTeX goes through several stages:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eReads the \u003ccode\u003e.tex\u003c/code\u003e file\u003c/strong\u003e — the compiler processes the preamble (\u003ccode\u003e\\usepackage{}\u003c/code\u003e) and the main document body.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBuilds document structure\u003c/strong\u003e — generates the table of contents, labels, references, and indexes.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eProcesses math symbols, images, and tables.\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCreates auxiliary files\u003c/strong\u003e such as:\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e.aux\u003c/code\u003e → for cross-references (\u003ccode\u003e\\ref\u003c/code\u003e, \u003ccode\u003e\\cite\u003c/code\u003e)\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e.toc\u003c/code\u003e → for the table of contents\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e.log\u003c/code\u003e → for error logs\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eProduces the final output (\u003ccode\u003e.pdf\u003c/code\u003e or \u003ccode\u003e.dvi\u003c/code\u003e).\u003c/strong\u003e\nBecause of these auxiliary files, \u003cstrong\u003eLaTeX documents often need to be compiled more than once\u003c/strong\u003e, especially when using cross-references, tables of contents, or bibliographies.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"compilation-via-terminal\" class=\"headerLink\"\u003e\n \u003ca href=\"#compilation-via-terminal\" class=\"header-mark\" aria-label=\"Direct link to section: Compilation via Terminal\"\u003e\u003c/a\u003eCompilation via Terminal\u003c/h2\u003e\u003cp\u003eFor local system users (Windows, macOS, Linux), compilation can be performed via the command line.\nExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003epdflatex document.tex\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ebibtex document\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003epdflatex document.tex\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003epdflatex document.tex\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eSteps:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eFirst compilation: generates \u003ccode\u003e.aux\u003c/code\u003e and \u003ccode\u003e.toc\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eRun \u003ccode\u003ebibtex\u003c/code\u003e (if a bibliography is used).\u003c/li\u003e\n\u003cli\u003eRecompile to update references.\u003c/li\u003e\n\u003cli\u003eFinal compilation for the finished output.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cblockquote\u003e\n \u003cp\u003e⚙️ If you’re using \u003ccode\u003ebiber\u003c/code\u003e (with \u003ccode\u003ebiblatex\u003c/code\u003e), replace \u003ccode\u003ebibtex\u003c/code\u003e with \u003ccode\u003ebiber document\u003c/code\u003e.\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"compilation-via-editor-environments\" class=\"headerLink\"\u003e\n \u003ca href=\"#compilation-via-editor-environments\" class=\"header-mark\" aria-label=\"Direct link to section: Compilation via Editor Environments\"\u003e\u003c/a\u003eCompilation via Editor Environments\u003c/h2\u003e\u003cp\u003eBesides the terminal, compilation can also be done directly from LaTeX editors. Common options include:\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/compile-document-in-latex/overleaf.webp\" title=\"overleaf\" data-thumbnail=\"/compile-document-in-latex/overleaf.webp\" data-sub-html=\"\u003ch2\u003eOverleaf interface\u003c/h2\u003e\u003cp\u003eoverleaf\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/compile-document-in-latex/overleaf.webp' srcset=\"/compile-document-in-latex/overleaf_hu_dfb4b0d794793c39.webp 800w, /compile-document-in-latex/overleaf_hu_1c78a052e03a3014.webp 1200w, /compile-document-in-latex/overleaf_hu_7cc483400809df4.webp 1600w\" alt=\"overleaf\" height=\"812\" width=\"1854\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eOverleaf interface\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch3 id=\"overleaf-online\" class=\"headerLink\"\u003e\n \u003ca href=\"#overleaf-online\" class=\"header-mark\" aria-label=\"Direct link to section: Overleaf (Online)\"\u003e\u003c/a\u003eOverleaf (Online)\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eAutomatically compiles whenever you save.\u003c/li\u003e\n\u003cli\u003eChoose the compiler via \u003cstrong\u003eMenu → Compiler → pdfLaTeX/XeLaTeX\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eNo local installation required.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"texworks--texstudio-offline\" class=\"headerLink\"\u003e\n \u003ca href=\"#texworks--texstudio-offline\" class=\"header-mark\" aria-label=\"Direct link to section: TeXworks / TeXstudio (Offline)\"\u003e\u003c/a\u003eTeXworks / TeXstudio (Offline)\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003ePress \u003cstrong\u003eCtrl + T\u003c/strong\u003e or click the “Typeset” icon.\u003c/li\u003e\n\u003cli\u003eSelect the compiler from the dropdown list.\u003c/li\u003e\n\u003cli\u003eSuitable for Windows/macOS users.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/compile-document-in-latex/texstudio.webp\" title=\"overleaf\" data-thumbnail=\"/compile-document-in-latex/texstudio.webp\" data-sub-html=\"\u003ch2\u003eTeXstudio interface\u003c/h2\u003e\u003cp\u003eoverleaf\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/compile-document-in-latex/texstudio.webp' srcset=\"/compile-document-in-latex/texstudio_hu_c8f13cd89399983f.webp 800w, /compile-document-in-latex/texstudio_hu_25eb52e74f6da60c.webp 1200w, /compile-document-in-latex/texstudio_hu_30a4a339c9a7753d.webp 1600w\" alt=\"overleaf\" height=\"889\" width=\"1909\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eTeXstudio interface\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"visual-studio-code--latex-workshop\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-studio-code--latex-workshop\" class=\"header-mark\" aria-label=\"Direct link to section: Visual Studio Code \u0026#43; LaTeX Workshop\"\u003e\u003c/a\u003eVisual Studio Code + LaTeX Workshop\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eA popular extension among modern users.\u003c/li\u003e\n\u003cli\u003eSupports build chain automation (e.g., pdflatex → bibtex → pdflatex → pdflatex).\u003c/li\u003e\n\u003cli\u003eConfiguration file: \u003ccode\u003e.vscode/settings.json\u003c/code\u003e\nExample configuration:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ejson\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s2\"\u003e\u0026#34;latex-workshop.latex.recipes\u0026#34;\u003c/span\u003e\u003cspan class=\"err\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nt\"\u003e\u0026#34;name\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;pdflatex -\u0026gt; bibtex -\u0026gt; pdflatex x2\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nt\"\u003e\u0026#34;tools\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;pdflatex\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;bibtex\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;pdflatex\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;pdflatex\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"common-compilation-errors\" class=\"headerLink\"\u003e\n \u003ca href=\"#common-compilation-errors\" class=\"header-mark\" aria-label=\"Direct link to section: Common Compilation Errors\"\u003e\u003c/a\u003eCommon Compilation Errors\u003c/h2\u003e\u003cp\u003eHere are common LaTeX compilation errors and how to fix them:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eError Message\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eCause\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eSolution\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e! Undefined control sequence.\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eUnknown command\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eEnsure the required package is loaded (\u003ccode\u003e\\usepackage{}\u003c/code\u003e)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e! LaTeX Error: File 'xxx.sty' not found.\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMissing package\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eInstall using a package manager (\u003ccode\u003etlmgr install xxx\u003c/code\u003e)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e! Missing $ inserted.\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMath mode error\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eWrap math expressions with \u003ccode\u003e$...$\u003c/code\u003e or \u003ccode\u003e\\[...\\]\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ePackage hyperref Warning: Token not allowed in a PDF string\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eInvalid characters in hyperlink\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eUse \u003ccode\u003e{}\u003c/code\u003e or replace special characters\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eOutput incomplete / references missing\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNot enough compilation runs\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCompile 2–3 times\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cblockquote\u003e\n \u003cp\u003e🧩 Tip: Check the \u003ccode\u003edocument.log\u003c/code\u003e file for detailed error messages and line numbers.\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"modern-workflow-and-automation\" class=\"headerLink\"\u003e\n \u003ca href=\"#modern-workflow-and-automation\" class=\"header-mark\" aria-label=\"Direct link to section: Modern Workflow and Automation\"\u003e\u003c/a\u003eModern Workflow and Automation\u003c/h2\u003e\u003cp\u003eFor efficiency, many users rely on \u003cstrong\u003eautomated compilation tools\u003c/strong\u003e, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003elatexmk\u003c/strong\u003e — automatically performs all compilation stages (LaTeX, BibTeX, index).\nExample:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003elatexmk -pdf document.tex\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003earara\u003c/strong\u003e — build system based on commands in \u003ccode\u003e.tex\u003c/code\u003e file comments.\nExample preamble:\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% arara: pdflatex\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% arara: bibtex\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% arara: pdflatex\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e arara: pdflatex\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMakefile\u003c/strong\u003e — for large projects with multiple \u003ccode\u003e.tex\u003c/code\u003e files.\nWith an automated workflow, you only need one command to trigger the entire compilation sequence.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"whether-using-overleaf-vs-code-or-the-terminal-the-principle-remains-the-same-make-sure-all-packages-references-and-resources-are-ready-before-the-final-compilation\" class=\"headerLink\"\u003e\n \u003ca href=\"#whether-using-overleaf-vs-code-or-the-terminal-the-principle-remains-the-same-make-sure-all-packages-references-and-resources-are-ready-before-the-final-compilation\" class=\"header-mark\" aria-label=\"Direct link to section: Compilation is the core of the LaTeX system — the process that transforms source code into a high-quality document. Understanding how compilers work, the build sequence, and error handling will help you write scientific documents more efficiently and professionally. Whether using Overleaf, VS Code, or the terminal, the principle remains the same: make sure all packages, references, and resources are ready before the final compilation.\"\u003e\u003c/a\u003eCompilation is the core of the LaTeX system — the process that transforms source code into a high-quality document.\nUnderstanding how compilers work, the build sequence, and error handling will help you write scientific documents more efficiently and professionally.\nWhether using Overleaf, VS Code, or the terminal, the principle remains the same: \u003cstrong\u003emake sure all packages, references, and resources are ready before the final compilation.\u003c/strong\u003e\u003c/h2\u003e\u003ch3 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eThe LaTeX Project. \u003ca href=\"https://www.latex-project.org\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.latex-project.org\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eOverleaf Documentation – \u003cem\u003eUnderstanding PDF Compilation\u003c/em\u003e\n\u003ca href=\"https://www.overleaf.com/learn/latex\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.overleaf.com/learn/latex\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTeX Users Group (TUG). \u003cem\u003eLaTeX Compilation Workflow\u003c/em\u003e.\n\u003ca href=\"https://tug.org\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://tug.org\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eWikibooks. \u003cem\u003eLaTeX/Installing and Compiling\u003c/em\u003e.\n\u003ca href=\"https://en.wikibooks.org/wiki/LaTeX/Installing_and_Compiling\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://en.wikibooks.org/wiki/LaTeX/Installing_and_Compiling\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e", "language": "en" }, { "title" : "Understanding Packages in LaTeX", "date_published" : "2025-10-17T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/latex-packages/", "url" : "https://www.aantriono.com/en/latex-packages/", "summary": "This article discusses the concepts, types, and functions of packages in LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the concepts, types, and functions of packages in LaTeX.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"hahahugoshortcode223s0hbhb\" class=\"headerLink\"\u003e\n \u003ca href=\"#hahahugoshortcode223s0hbhb\" class=\"header-mark\" aria-label=\"Direct link to section: \u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eLaTeX is a typesetting system that has become the de facto standard in the academic and scientific research world. One of LaTeX\u0026rsquo;s main advantages is its flexibility through the use of \u003cstrong\u003epackages\u003c/strong\u003e — additional modules that extend the core system\u0026rsquo;s capabilities. This article comprehensively discusses the concepts, functions, and management of packages in LaTeX, complete with real-world application examples for academic documents such as theses and dissertations. Additionally, this article highlights common issues like package conflicts and provides practical solutions to address them.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\"\u003e\u003c/a\u003e\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eLaTeX is a typesetting system that has become the de facto standard in the academic and scientific research world. One of LaTeX\u0026rsquo;s main advantages is its flexibility through the use of \u003cstrong\u003epackages\u003c/strong\u003e — additional modules that extend the core system\u0026rsquo;s capabilities. This article comprehensively discusses the concepts, functions, and management of packages in LaTeX, complete with real-world application examples for academic documents such as theses and dissertations. Additionally, this article highlights common issues like package conflicts and provides practical solutions to address them.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/h2\u003e\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"according-to-wayne-state-university-libraries-2024-latex-packages-are-extension-components-that-facilitate-users-in-managing-elements-such-as-mathematics-graphics-bibliographic-references-and-page-layout-without-these-packages-users-would-have-to-write-code-from-scratch-for-every-desired-feature--something-inefficient-and-prone-to-errors\" class=\"headerLink\"\u003e\n \u003ca href=\"#according-to-wayne-state-university-libraries-2024-latex-packages-are-extension-components-that-facilitate-users-in-managing-elements-such-as-mathematics-graphics-bibliographic-references-and-page-layout-without-these-packages-users-would-have-to-write-code-from-scratch-for-every-desired-feature--something-inefficient-and-prone-to-errors\" class=\"header-mark\" aria-label=\"Direct link to section: In the world of scientific writing, LaTeX is known for its ability to produce precise, consistent, and aesthetic documents. However, the true power of LaTeX lies not only in its core engine but in the thousands of additional packages that provide advanced functionality. According to Wayne State University Libraries (2024), LaTeX packages are extension components that facilitate users in managing elements such as mathematics, graphics, bibliographic references, and page layout. Without these packages, users would have to write code from scratch for every desired feature — something inefficient and prone to errors.\"\u003e\u003c/a\u003eIn the world of scientific writing, LaTeX is known for its ability to produce precise, consistent, and aesthetic documents. However, the true power of LaTeX lies not only in its core engine but in the \u003cstrong\u003ethousands of additional packages\u003c/strong\u003e that provide advanced functionality.\nAccording to \u003cem\u003eWayne State University Libraries (2024)\u003c/em\u003e, LaTeX packages are extension components that facilitate users in managing elements such as mathematics, graphics, bibliographic references, and page layout. Without these packages, users would have to write code from scratch for every desired feature — something inefficient and prone to errors.\u003c/h2\u003e\u003ch2 id=\"basic-concepts-of-latex-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-concepts-of-latex-packages\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Concepts of LaTeX Packages\"\u003e\u003c/a\u003eBasic Concepts of LaTeX Packages\u003c/h2\u003e\u003ch3 id=\"what-is-a-package\" class=\"headerLink\"\u003e\n \u003ca href=\"#what-is-a-package\" class=\"header-mark\" aria-label=\"Direct link to section: What is a Package?\"\u003e\u003c/a\u003eWhat is a Package?\u003c/h3\u003e\u003cp\u003eA package is a collection of additional macros and commands that extend LaTeX\u0026rsquo;s basic functions. Package files typically have the \u003ccode\u003e.sty\u003c/code\u003e extension and are activated using the command:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[options]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epackage-name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis command is written \u003cstrong\u003ein the preamble section\u003c/strong\u003e, the area before \u003ccode\u003e\\begin{document}\u003c/code\u003e. Example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-16\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=1in]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"ctan\" class=\"headerLink\"\u003e\n \u003ca href=\"#ctan\" class=\"header-mark\" aria-label=\"Direct link to section: Each package has its own documentation that can be found on CTAN (Comprehensive TeX Archive Network), the primary source for LaTeX packages worldwide. CTAN Home Page: Comprehensive TeX Archive Network \"\u003e\u003c/a\u003eEach package has its own documentation that can be found on \u003cstrong\u003eCTAN (Comprehensive TeX Archive Network)\u003c/strong\u003e, the primary source for LaTeX packages worldwide.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/latex-packages/ctan.webp\" title=\"ctan\" data-thumbnail=\"/latex-packages/ctan.webp\" data-sub-html=\"\u003ch2\u003eCTAN Home Page: Comprehensive TeX Archive Network\u003c/h2\u003e\u003cp\u003ectan\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/latex-packages/ctan.webp' srcset=\"/latex-packages/ctan_hu_9a132a94deef6991.webp 800w, /latex-packages/ctan_hu_c57ef231a8af710e.webp 1200w, /latex-packages/ctan_hu_d30c875474f51d6c.webp 1600w\" alt=\"ctan\" height=\"808\" width=\"1794\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCTAN Home Page: Comprehensive TeX Archive Network\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/h2\u003e\u003ch2 id=\"types-and-functions-of-popular-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-and-functions-of-popular-packages\" class=\"header-mark\" aria-label=\"Direct link to section: Types and Functions of Popular Packages\"\u003e\u003c/a\u003eTypes and Functions of Popular Packages\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003ePackage Name\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eMain Function\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eExample Usage\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eamsmath\u003c/strong\u003e, \u003cstrong\u003eamssymb\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eProvides advanced mathematical symbols and environments\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eWriting complex equations, matrices, integrals\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003egraphicx\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eInserts external images (PNG, JPG, PDF)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\includegraphics[width=0.5\\textwidth]{image.png}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003egeometry\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdjusts margins and page size\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\usepackage[margin=1in]{geometry}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003ehyperref\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdds hyperlinks and PDF metadata\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\usepackage[colorlinks=true]{hyperref}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003ebiblatex\u003c/strong\u003e / \u003cstrong\u003enatbib\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eManages automatic citations and bibliography\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\usepackage[backend=biber,style=apa]{biblatex}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003etikz\u003c/strong\u003e, \u003cstrong\u003epgfplots\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCreates vector graphics and diagrams directly in LaTeX\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eScientific diagrams, workflows, data graphs\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003ecleveref\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eSimplifies cross-referencing\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\cref{label}\u003c/code\u003e for automatic references\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003excolor\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eControls text colors and page elements\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eText highlighting or colored tables\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"case-study-1-preamble-for-thesis-or-dissertation\" class=\"headerLink\"\u003e\n \u003ca href=\"#case-study-1-preamble-for-thesis-or-dissertation\" class=\"header-mark\" aria-label=\"Direct link to section: Case Study 1: Preamble for Thesis or Dissertation\"\u003e\u003c/a\u003eCase Study 1: Preamble for Thesis or Dissertation\u003c/h2\u003e\u003cp\u003eThe preamble is the most crucial part of an academic document as it determines page format, font type, and citation style. Here\u0026rsquo;s an example preamble commonly used in university academic works:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-17\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[12pt,a4paper]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ereport\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Language and encoding\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[utf8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003einputenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[T1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efontenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[english]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Page layout\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=3cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Mathematics and science packages\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath, amssymb, amsthm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esiunitx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Graphics and tables\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebooktabs\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecaption\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esubcaption\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Citations and bibliography\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[backend=biber,style=apa]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebiblatex\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\addbibresource\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebibliography.bib\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Hyperlinks and PDF metadata\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[hidelinks]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ehyperref\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hypersetup\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e pdfauthor=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAuthor Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e pdftitle=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eThesis or Dissertation Title\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e pdfsubject=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAcademic Document\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e pdfkeywords=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eLaTeX, Thesis, Packages, Academic\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"use-a-modular-approach-by-storing-the-preamble-section-in-a-separate-file-preambletex-for-easy-maintenance-and-reuse-across-various-projects\" class=\"headerLink\"\u003e\n \u003ca href=\"#use-a-modular-approach-by-storing-the-preamble-section-in-a-separate-file-preambletex-for-easy-maintenance-and-reuse-across-various-projects\" class=\"header-mark\" aria-label=\"Direct link to section: 💡 Tips: Use a modular approach by storing the preamble section in a separate file (preamble.tex) for easy maintenance and reuse across various projects.\"\u003e\u003c/a\u003e💡 \u003cem\u003eTips:\u003c/em\u003e\nUse a modular approach by storing the preamble section in a separate file (\u003ccode\u003epreamble.tex\u003c/code\u003e) for easy maintenance and reuse across various projects.\u003c/h2\u003e\u003ch2 id=\"case-study-2-resolving-conflicts-between-hyperref-and-geometry-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#case-study-2-resolving-conflicts-between-hyperref-and-geometry-packages\" class=\"header-mark\" aria-label=\"Direct link to section: Case Study 2: Resolving Conflicts Between hyperref and geometry Packages\"\u003e\u003c/a\u003eCase Study 2: Resolving Conflicts Between \u003ccode\u003ehyperref\u003c/code\u003e and \u003ccode\u003egeometry\u003c/code\u003e Packages\u003c/h2\u003e\u003cp\u003eOne of the most common issues in LaTeX documents is \u003cstrong\u003epackage conflicts\u003c/strong\u003e, especially between \u003ccode\u003ehyperref\u003c/code\u003e and \u003ccode\u003egeometry\u003c/code\u003e. Both packages often manage page layout elements and PDF metadata, which can cause error messages like:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-18\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nx\"\u003eOption\u003c/span\u003e \u003cspan class=\"nx\"\u003eclash\u003c/span\u003e \u003cspan class=\"k\"\u003efor\u003c/span\u003e \u003cspan class=\"kn\"\u003epackage\u003c/span\u003e \u003cspan class=\"nx\"\u003egeometry\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution:\"\u003e\u003c/a\u003eSolution:\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eCall \u003ccode\u003egeometry\u003c/code\u003e before \u003ccode\u003ehyperref\u003c/code\u003e\u003c/strong\u003e, because \u003ccode\u003ehyperref\u003c/code\u003e reads existing layout settings.\u003c/li\u003e\n\u003cli\u003eUse configuration like this:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-19\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=1in]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[colorlinks=true, linkcolor=blue]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ehyperref\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eAvoid calling \u003ccode\u003egeometry\u003c/code\u003e again after \u003ccode\u003ehyperref\u003c/code\u003e. If you need to change margins mid-document, use:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-20\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\newgeometry\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emargin=2cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-way-option-conflicts-will-be-avoided-and-hyperlinks-will-function-properly-without-changing-the-page-format\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-way-option-conflicts-will-be-avoided-and-hyperlinks-will-function-properly-without-changing-the-page-format\" class=\"header-mark\" aria-label=\"Direct link to section: This way, option conflicts will be avoided and hyperlinks will function properly without changing the page format.\"\u003e\u003c/a\u003eThis way, option conflicts will be avoided and hyperlinks will function properly without changing the page format.\u003c/h2\u003e\u003ch2 id=\"recommended-best-practices\" class=\"headerLink\"\u003e\n \u003ca href=\"#recommended-best-practices\" class=\"header-mark\" aria-label=\"Direct link to section: Recommended Best Practices\"\u003e\u003c/a\u003eRecommended Best Practices\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eUse only necessary packages\u003c/strong\u003e to maintain document stability.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eUpdate LaTeX distribution\u003c/strong\u003e (TeX Live, MiKTeX) regularly to get the latest package versions.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eRead official documentation\u003c/strong\u003e for each package on CTAN before use.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePay attention to loading order\u003c/strong\u003e, especially for packages that modify page appearance or references.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eUse comments\u003c/strong\u003e in the preamble to explain each package\u0026rsquo;s function for future understanding.\nExample of good commenting:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-21\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Packages for advanced mathematics formatting\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath, amssymb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"latex-packages-are-the-backbone-of-the-latex-systems-flexibility-by-understanding-how-they-work-their-loading-structure-and-potential-inter-package-conflicts-users-can-create-neat-professional-and-efficient-academic-documents-whether-for-theses-dissertations-or-scientific-publications-mastering-packages-is-a-crucial-step-toward-optimal-productivity-in-the-digital-academic-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-packages-are-the-backbone-of-the-latex-systems-flexibility-by-understanding-how-they-work-their-loading-structure-and-potential-inter-package-conflicts-users-can-create-neat-professional-and-efficient-academic-documents-whether-for-theses-dissertations-or-scientific-publications-mastering-packages-is-a-crucial-step-toward-optimal-productivity-in-the-digital-academic-environment\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX packages are the backbone of the LaTeX system\u0026rsquo;s flexibility. By understanding how they work, their loading structure, and potential inter-package conflicts, users can create neat, professional, and efficient academic documents. Whether for theses, dissertations, or scientific publications, mastering packages is a crucial step toward optimal productivity in the digital academic environment.\"\u003e\u003c/a\u003eLaTeX packages are the backbone of the LaTeX system\u0026rsquo;s flexibility. By understanding how they work, their loading structure, and potential inter-package conflicts, users can create neat, professional, and efficient academic documents. Whether for theses, dissertations, or scientific publications, mastering packages is a crucial step toward optimal productivity in the digital academic environment.\u003c/h2\u003e\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eWayne State University Libraries. \u003cem\u003eHow to Use LaTeX: Packages.\u003c/em\u003e\n\u003ca href=\"https://guides.lib.wayne.edu/latex/packages\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://guides.lib.wayne.edu/latex/packages\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eThe LaTeX Project. \u003cem\u003eCTAN: Comprehensive TeX Archive Network.\u003c/em\u003e\n\u003ca href=\"https://ctan.org\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://ctan.org\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eOverleaf Documentation. \u003cem\u003eLaTeX Packages and Configuration.\u003c/em\u003e\n\u003ca href=\"https://www.overleaf.com/learn\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.overleaf.com/learn\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eStackExchange LaTeX Community. \u003cem\u003eCommon Package Conflicts and Solutions.\u003c/em\u003e\n\u003ca href=\"https://tex.stackexchange.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://tex.stackexchange.com\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e", "language": "en" }, { "title" : "Debian: The Universal Operating System", "date_published" : "2025-10-16T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/debian-linux/", "url" : "https://www.aantriono.com/en/debian-linux/", "summary": "Debian is one of the oldest and most respected Linux distributions, known for its stability, security, and free software principles. In this article, we will discuss its history, characteristics, variants, use in software development, as well as its advantages and disadvantages — in depth yet easy to understand.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eDebian is one of the oldest and most respected Linux distributions, known for its stability, security, and free software principles.\nIn this article, we will discuss its history, characteristics, variants, use in software development, as well as its advantages and disadvantages — in depth yet easy to understand.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"a-brief-history-of-debian\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-brief-history-of-debian\" class=\"header-mark\" aria-label=\"Direct link to section: A Brief History of Debian\"\u003e\u003c/a\u003eA Brief History of Debian\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eDebian was founded in August 1993 by \u003cstrong\u003eIan Murdock\u003c/strong\u003e as a community project aimed at providing a \u003cem\u003euniversal, free, and high-quality\u003c/em\u003e Linux operating system.\u003c/li\u003e\n\u003cli\u003eFrom the beginning, Debian has embraced the spirit of free software and global collaboration.\u003c/li\u003e\n\u003cli\u003eDebian release names follow characters from the \u003cem\u003eToy Story\u003c/em\u003e movies (such as \u0026ldquo;Jessie\u0026rdquo;, \u0026ldquo;Bullseye\u0026rdquo;, \u0026ldquo;Bookworm\u0026rdquo;, and so on).\u003c/li\u003e\n\u003cli\u003eDebian serves as the foundation for many other popular distributions such as \u003cstrong\u003eUbuntu\u003c/strong\u003e, \u003cstrong\u003eLinux Mint\u003c/strong\u003e, and other variants.\u003c/li\u003e\n\u003cli\u003eAnalysis of Debian 3.0 showed that this project was massive in code scope: over 105 million lines of source code.\n\u003cblockquote\u003e\n \u003cp\u003eSource: \u003ca href=\"https://arxiv.org/abs/cs/0506067\" target=\"_blank\" rel=\"noopener noreferrer\"\u003earxiv.org\u003c/a\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"philosophy-and-community\" class=\"headerLink\"\u003e\n \u003ca href=\"#philosophy-and-community\" class=\"header-mark\" aria-label=\"Direct link to section: Philosophy and Community\"\u003e\u003c/a\u003ePhilosophy and Community\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/debian-linux/debian.webp\" title=\"Debian\" data-thumbnail=\"/debian-linux/debian.webp\" data-sub-html=\"\u003ch2\u003eDebian.org website homepage\u003c/h2\u003e\u003cp\u003eDebian\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/debian-linux/debian.webp' srcset=\"/debian-linux/debian_hu_86c09aadffc7b191.webp 800w, /debian-linux/debian_hu_d1d732edc0e692a2.webp 1200w, /debian-linux/debian_hu_5abbb2def56afd.webp 1600w\" alt=\"Debian\" height=\"788\" width=\"1857\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eDebian.org website homepage\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch3 id=\"debian-philosophy\" class=\"headerLink\"\u003e\n \u003ca href=\"#debian-philosophy\" class=\"header-mark\" aria-label=\"Direct link to section: Debian Philosophy\"\u003e\u003c/a\u003eDebian Philosophy\u003c/h3\u003e\u003cp\u003eDebian is based on the following principles:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eSoftware freedom\u003c/strong\u003e — only using and supporting free software.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eQuality and stability\u003c/strong\u003e — every package is rigorously tested before entering stable releases.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTransparency and collaboration\u003c/strong\u003e — decisions are made through community consensus.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eUser focus\u003c/strong\u003e — technical decisions are directed toward the interests of users and developers.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"debian-community\" class=\"headerLink\"\u003e\n \u003ca href=\"#debian-community\" class=\"header-mark\" aria-label=\"Direct link to section: Debian Community\"\u003e\u003c/a\u003eDebian Community\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eFully managed by volunteers from around the world.\u003c/li\u003e\n\u003cli\u003eHas dedicated teams such as maintainers, security teams, and documentation teams.\u003c/li\u003e\n\u003cli\u003eMajor decisions are made through community voting processes.\u003c/li\u003e\n\u003cli\u003eThe Debian community is also known for its extremely comprehensive documentation.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"debian-releases--release-model\" class=\"headerLink\"\u003e\n \u003ca href=\"#debian-releases--release-model\" class=\"header-mark\" aria-label=\"Direct link to section: Debian Releases \u0026amp; Release Model\"\u003e\u003c/a\u003eDebian Releases \u0026amp; Release Model\u003c/h2\u003e\u003cp\u003eDebian has three main branches:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eBranch\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eCharacteristics\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003ePrimary Use\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eStable\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eOfficial and very stable version\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eProduction systems and servers\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eTesting\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNewer packages, being tested\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eUsers who want a balance of stability and updates\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eUnstable (Sid)\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLatest and experimental packages\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eDevelopers and testers\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eDebian stable releases appear every 2–3 years.\u003c/li\u003e\n\u003cli\u003eNew packages are tested in \u003cem\u003eunstable\u003c/em\u003e, then move to \u003cem\u003etesting\u003c/em\u003e, and finally to \u003cem\u003estable\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eUsers can also use \u003cem\u003ebackports\u003c/em\u003e to get newer package versions without changing releases.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"architecture-and-package-management\" class=\"headerLink\"\u003e\n \u003ca href=\"#architecture-and-package-management\" class=\"header-mark\" aria-label=\"Direct link to section: Architecture and Package Management\"\u003e\u003c/a\u003eArchitecture and Package Management\u003c/h2\u003e\u003ch3 id=\"supported-architectures\" class=\"headerLink\"\u003e\n \u003ca href=\"#supported-architectures\" class=\"header-mark\" aria-label=\"Direct link to section: Supported Architectures\"\u003e\u003c/a\u003eSupported Architectures\u003c/h3\u003e\u003cp\u003eDebian supports various hardware architectures such as \u003cstrong\u003eamd64\u003c/strong\u003e, \u003cstrong\u003ei386\u003c/strong\u003e, and \u003cstrong\u003eARM\u003c/strong\u003e (various variants).\u003c/p\u003e\n\u003ch3 id=\"package-system--management\" class=\"headerLink\"\u003e\n \u003ca href=\"#package-system--management\" class=\"header-mark\" aria-label=\"Direct link to section: Package System \u0026amp; Management\"\u003e\u003c/a\u003ePackage System \u0026amp; Management\u003c/h3\u003e\u003cp\u003eDebian uses the \u003cstrong\u003e.deb\u003c/strong\u003e package system with management through \u003cstrong\u003eAPT (Advanced Package Tool)\u003c/strong\u003e.\nSome basic commands:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt update\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt upgrade\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt install \u0026lt;package-name\u0026gt;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt remove \u0026lt;package-name\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eRepositories are divided into three sections:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003emain\u003c/strong\u003e — completely free software.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003econtrib\u003c/strong\u003e — free software that depends on non-free packages.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003enon-free\u003c/strong\u003e — software that is not completely license-free.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"debian-variants-and-derivatives\" class=\"headerLink\"\u003e\n \u003ca href=\"#debian-variants-and-derivatives\" class=\"header-mark\" aria-label=\"Direct link to section: Debian Variants and Derivatives\"\u003e\u003c/a\u003eDebian Variants and Derivatives\u003c/h2\u003e\u003cp\u003eDue to its open nature, many distributions are built on top of Debian. Here are some examples:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eUbuntu\u003c/strong\u003e — the most popular Debian derivative with a user-friendly interface.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLinux Mint Debian Edition (LMDE)\u003c/strong\u003e — Mint version based directly on Debian.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSolydXK\u003c/strong\u003e — based on Debian Stable, with KDE and Xfce desktops.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eElive\u003c/strong\u003e — uses the Enlightenment desktop with an elegant appearance.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAVLinux\u003c/strong\u003e — focused on audio and multimedia production.\nThese derivatives typically add configurations, custom kernels, or additional packages according to user needs.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"debian-for-developers--programmers\" class=\"headerLink\"\u003e\n \u003ca href=\"#debian-for-developers--programmers\" class=\"header-mark\" aria-label=\"Direct link to section: Debian for Developers / Programmers\"\u003e\u003c/a\u003eDebian for Developers / Programmers\u003c/h2\u003e\u003cp\u003eDebian is widely used by software developers due to its stability and package availability.\u003c/p\u003e\n\u003ch3 id=\"development-tools--environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#development-tools--environment\" class=\"header-mark\" aria-label=\"Direct link to section: Development Tools \u0026amp; Environment\"\u003e\u003c/a\u003eDevelopment Tools \u0026amp; Environment\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ebuild-essential\u003c/strong\u003e package: gcc, g++, make, and basic libraries.\u003c/li\u003e\n\u003cli\u003ePopular programming languages such as \u003cstrong\u003ePython\u003c/strong\u003e, \u003cstrong\u003eNode.js\u003c/strong\u003e, \u003cstrong\u003eJava\u003c/strong\u003e, and \u003cstrong\u003eGo\u003c/strong\u003e are available directly from repositories.\u003c/li\u003e\n\u003cli\u003eVersion control systems like \u003cstrong\u003eGit\u003c/strong\u003e are easily installed.\u003c/li\u003e\n\u003cli\u003ePopular editors like \u003cstrong\u003eVS Code\u003c/strong\u003e and \u003cstrong\u003eSublime Text\u003c/strong\u003e can be installed with \u003ccode\u003e.deb\u003c/code\u003e packages or external repositories.\n\u003cblockquote\u003e\n \u003cp\u003eReference: \u003ca href=\"https://dev.to/seanpetiya/a-simple-dev-environment-with-visual-studio-code-on-debian-linux-4mie\" target=\"_blank\" rel=\"noopener noreferrer\"\u003edev.to/seanpetiya\u003c/a\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"advantages-for-developers\" class=\"headerLink\"\u003e\n \u003ca href=\"#advantages-for-developers\" class=\"header-mark\" aria-label=\"Direct link to section: Advantages for Developers\"\u003e\u003c/a\u003eAdvantages for Developers\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eStable and secure\u003c/strong\u003e — suitable for long-term development.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eHigh reproducibility\u003c/strong\u003e — consistent configuration across systems.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLarge community\u003c/strong\u003e — comprehensive documentation and active community support.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFull customization\u003c/strong\u003e — users can determine package versions or build their own.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"challenges\" class=\"headerLink\"\u003e\n \u003ca href=\"#challenges\" class=\"header-mark\" aria-label=\"Direct link to section: Challenges\"\u003e\u003c/a\u003eChallenges\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003ePackage versions in \u003cem\u003estable\u003c/em\u003e are often behind the latest versions.\u003c/li\u003e\n\u003cli\u003eSome proprietary drivers require additional configuration.\u003c/li\u003e\n\u003cli\u003eFor new users, the CLI might feel difficult at first.\n\u003cblockquote\u003e\n \u003cp\u003e\u0026ldquo;Debian is great but you\u0026rsquo;re stuck with pretty old versions of many packages on Debian stable.\u0026rdquo;\n— \u003ca href=\"https://www.reddit.com/r/linuxquestions/comments/1ee4pmk/best_distro-for-programming-and-developing\" target=\"_blank\" rel=\"noopener noreferrer\"\u003er/linuxquestions\u003c/a\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"advantages-and-disadvantages-of-debian\" class=\"headerLink\"\u003e\n \u003ca href=\"#advantages-and-disadvantages-of-debian\" class=\"header-mark\" aria-label=\"Direct link to section: Advantages and Disadvantages of Debian\"\u003e\u003c/a\u003eAdvantages and Disadvantages of Debian\u003c/h2\u003e\u003ch3 id=\"advantages\" class=\"headerLink\"\u003e\n \u003ca href=\"#advantages\" class=\"header-mark\" aria-label=\"Direct link to section: Advantages\"\u003e\u003c/a\u003eAdvantages\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eVery stable and secure.\u003c/li\u003e\n\u003cli\u003eCompletely free and open source.\u003c/li\u003e\n\u003cli\u003eLarge community and extensive documentation.\u003c/li\u003e\n\u003cli\u003eSupports many architectures.\u003c/li\u003e\n\u003cli\u003eVery large package ecosystem.\u003c/li\u003e\n\u003cli\u003eFlexible for both servers and desktops.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"disadvantages\" class=\"headerLink\"\u003e\n \u003ca href=\"#disadvantages\" class=\"header-mark\" aria-label=\"Direct link to section: Disadvantages\"\u003e\u003c/a\u003eDisadvantages\u003c/h3\u003e\u003col\u003e\n\u003cli\u003ePackage versions in \u003cem\u003estable\u003c/em\u003e can be outdated.\u003c/li\u003e\n\u003cli\u003eInstalling non-free drivers is not always easy.\u003c/li\u003e\n\u003cli\u003eLearning curve is quite steep for beginners.\u003c/li\u003e\n\u003cli\u003eMajor updates require extra attention.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"tips-for-getting-started-with-debian\" class=\"headerLink\"\u003e\n \u003ca href=\"#tips-for-getting-started-with-debian\" class=\"header-mark\" aria-label=\"Direct link to section: Tips for Getting Started with Debian\"\u003e\u003c/a\u003eTips for Getting Started with Debian\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eUse the \u003cstrong\u003estable release\u003c/strong\u003e for beginners or servers.\u003c/li\u003e\n\u003cli\u003eEnable \u003cem\u003econtrib\u003c/em\u003e and \u003cem\u003enon-free\u003c/em\u003e repositories if needed.\u003c/li\u003e\n\u003cli\u003eBackup the \u003ccode\u003esources.list\u003c/code\u003e file before adding new repositories.\u003c/li\u003e\n\u003cli\u003eUse \u003ccode\u003eapt-mark hold \u0026lt;package\u0026gt;\u003c/code\u003e to hold specific versions.\u003c/li\u003e\n\u003cli\u003eLearn how to build \u003ccode\u003e.deb\u003c/code\u003e packages yourself.\u003c/li\u003e\n\u003cli\u003eFollow the Debian community through forums or mailing lists.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eDebian is an important pillar in the Linux world.\nThe combination of stability, security, and community spirit makes it a strong foundation for many other distributions.\nFor new users, Debian might seem technical. However, for developers and system administrators, Debian is a reliable, secure, and flexible choice — suitable for desktops, servers, and software development.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eAdditional References:\u003c/strong\u003e\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ca href=\"https://www.debian.org/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eDebian Official Website\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://en.wikipedia.org/wiki/Debian\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eWikipedia: Debian\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://wiki.debian.org/DevelopersCorner\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eDebian Developer\u0026rsquo;s Corner\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/blockquote\u003e", "language": "en" }, { "title" : "Text Formatting in LaTeX", "date_published" : "2025-10-16T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/text-format-in-latex/", "url" : "https://www.aantriono.com/en/text-format-in-latex/", "summary": "This article will cover various basic and advanced text formatting techniques in LaTeX, including the use of font styles, font sizes, text alignment, and other elements commonly used in LaTeX documents.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article will cover various basic and advanced text formatting techniques in LaTeX, including the use of font styles, font sizes, text alignment, and other elements commonly used in LaTeX documents.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"font-styles\" class=\"headerLink\"\u003e\n \u003ca href=\"#font-styles\" class=\"header-mark\" aria-label=\"Direct link to section: Font Styles\"\u003e\u003c/a\u003eFont Styles\u003c/h2\u003e\u003cp\u003eLaTeX provides various commands to change text font styles:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\textbf{...}\u003c/code\u003e: Makes text bold.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\textit{...}\u003c/code\u003e: Makes text italic.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\underline{...}\u003c/code\u003e: Underlines text.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\emph{...}\u003c/code\u003e: Emphasizes text, usually by making it italic.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\textsc{...}\u003c/code\u003e: Converts text to small caps.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\textrm{...}\u003c/code\u003e: Sets text using roman (serif) font.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\textsf{...}\u003c/code\u003e: Sets text using sans-serif font.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\texttt{...}\u003c/code\u003e: Sets text using monospace (typewriter) font.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\textnormal{...}\u003c/code\u003e: Returns text to default font style.\nUsage example:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eThis text is bold\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"k\"\u003e\\textit\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eitalic\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"k\"\u003e\\underline\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eunderlined\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"k\"\u003e\\emph\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eemphasized\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"k\"\u003e\\textsc\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esmall caps\u003cspan class=\"nb\"\u003e}\u003c/span\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"font-sizes\" class=\"headerLink\"\u003e\n \u003ca href=\"#font-sizes\" class=\"header-mark\" aria-label=\"Direct link to section: Font Sizes\"\u003e\u003c/a\u003eFont Sizes\u003c/h2\u003e\u003cp\u003eTo change font size, LaTeX provides several commands:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\tiny{...}\u003c/code\u003e: Very small font size.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\scriptsize{...}\u003c/code\u003e: Small font size for scripts.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\footnotesize{...}\u003c/code\u003e: Font size for footnotes.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\small{...}\u003c/code\u003e: Small font size.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\normalsize{...}\u003c/code\u003e: Normal (default) font size.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\large{...}\u003c/code\u003e: Large font size.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\Large{...}\u003c/code\u003e: Very large font size.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\huge{...}\u003c/code\u003e: Huge font size.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\Huge{...}\u003c/code\u003e: Largest font size.\nUsage example:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\huge\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eThis text is very large\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"text-alignment\" class=\"headerLink\"\u003e\n \u003ca href=\"#text-alignment\" class=\"header-mark\" aria-label=\"Direct link to section: Text Alignment\"\u003e\u003c/a\u003eText Alignment\u003c/h2\u003e\u003cp\u003eLaTeX allows text alignment with several commands:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\centering\u003c/code\u003e: Centers text.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\raggedright\u003c/code\u003e: Aligns text to the left.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\raggedleft\u003c/code\u003e: Aligns text to the right.\nUsage example:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Large\u003c/span\u003e This text is centered\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"lists-and-bullet-points\" class=\"headerLink\"\u003e\n \u003ca href=\"#lists-and-bullet-points\" class=\"header-mark\" aria-label=\"Direct link to section: Lists and Bullet Points\"\u003e\u003c/a\u003eLists and Bullet Points\u003c/h2\u003e\u003cp\u003eLaTeX provides environments for creating lists:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eUnordered list:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eitemize\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e First point\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Second point\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Third point\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eitemize\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eOrdered list:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e First item\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Second item\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Third item\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"colors-and-backgrounds\" class=\"headerLink\"\u003e\n \u003ca href=\"#colors-and-backgrounds\" class=\"header-mark\" aria-label=\"Direct link to section: Colors and Backgrounds\"\u003e\u003c/a\u003eColors and Backgrounds\u003c/h2\u003e\u003cp\u003eTo add color to text or backgrounds, you need to load the \u003ccode\u003excolor\u003c/code\u003e package:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eUsage example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ered\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eThis text is red\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTo add a background to text:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eyellow\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eText with yellow background\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"tables-and-figures\" class=\"headerLink\"\u003e\n \u003ca href=\"#tables-and-figures\" class=\"header-mark\" aria-label=\"Direct link to section: Tables and Figures\"\u003e\u003c/a\u003eTables and Figures\u003c/h2\u003e\u003cp\u003eLaTeX allows easy creation of tables and insertion of images:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eTables:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|c|c|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eColumn 1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Column 2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Column 3 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eData 1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Data 2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Data 3 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eFigures:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.5\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.jpg\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eImage description\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"using-macros\" class=\"headerLink\"\u003e\n \u003ca href=\"#using-macros\" class=\"header-mark\" aria-label=\"Direct link to section: Using Macros\"\u003e\u003c/a\u003eUsing Macros\u003c/h2\u003e\u003cp\u003eMacros in LaTeX allow you to define new commands to simplify writing:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\myname\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eJohn Doe\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen, you can use \u003ccode\u003e\\myname\u003c/code\u003e throughout the document to insert \u0026ldquo;John Doe\u0026rdquo;.\u003c/p\u003e\n\u003ch2 id=\"special-characters\" class=\"headerLink\"\u003e\n \u003ca href=\"#special-characters\" class=\"header-mark\" aria-label=\"Direct link to section: Special Characters\"\u003e\u003c/a\u003eSpecial Characters\u003c/h2\u003e\u003cp\u003eLaTeX has several special characters with specific functions, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e# $ % ^ \u0026amp; _ { } ~ _\u003c/code\u003e and \u003ccode\u003e\\\u003c/code\u003e.\nTo use these characters as regular text, you need to use special commands:\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\#\u003c/code\u003e for hash sign (\u003ccode\u003e#\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\%\u003c/code\u003e for percent (\u003ccode\u003e%\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\\u0026amp;\u003c/code\u003e for ampersand (\u003ccode\u003e\u0026amp;\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\_\u003c/code\u003e for underscore (\u003ccode\u003e_\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\{\u003c/code\u003e and \u003ccode\u003e\\}\u003c/code\u003e for curly braces (\u003ccode\u003e{\u003c/code\u003e and \u003ccode\u003e}\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\~{}\u003c/code\u003e for tilde (\u003ccode\u003e~\u003c/code\u003e).\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\textbackslash\u003c/code\u003e for backslash (\u003ccode\u003e\\\u003c/code\u003e).\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eLaTeX is a very powerful tool for formatting text in scientific and technical documents. By understanding and utilizing the basic and advanced commands discussed, you can create documents that are not only informative but also visually appealing. Continue exploring and experimenting with LaTeX to maximize its potential in document preparation.\u003c/p\u003e", "language": "en" }, { "title" : "LaTeX Document Structure", "date_published" : "2025-10-14T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/latex-document-structure/", "url" : "https://www.aantriono.com/en/latex-document-structure/", "summary": "This article discusses LaTeX document structure for letter, article, report, and book document classes.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses LaTeX document structure for letter, article, report, and book document classes.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cstrong\u003eLaTeX\u003c/strong\u003e is a markup-based typesetting system widely used for writing scientific works, reports, books, and theses.\nOne popular platform for writing and composing LaTeX documents is \u003cstrong\u003eOverleaf\u003c/strong\u003e, an online editor that enables real-time collaboration and provides complete documentation and templates.\nBefore starting document writing, it\u0026rsquo;s important to understand the basic structure of LaTeX documents.\nA \u003ccode\u003e.tex\u003c/code\u003e file generally consists of three main parts:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003ePreamble\u003c/strong\u003e – initial document settings before \u003ccode\u003e\\begin{document}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBody\u003c/strong\u003e – main document content between \u003ccode\u003e\\begin{document}\u003c/code\u003e and \u003ccode\u003e\\end{document}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAdditional sections\u003c/strong\u003e – such as appendices, bibliography, or index\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"general-structure-of-latex-documents\" class=\"headerLink\"\u003e\n \u003ca href=\"#general-structure-of-latex-documents\" class=\"header-mark\" aria-label=\"Direct link to section: General Structure of LaTeX Documents\"\u003e\u003c/a\u003eGeneral Structure of LaTeX Documents\u003c/h2\u003e\u003cp\u003eThe minimal structure of a LaTeX document is as follows:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[utf8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003einputenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eDocument Title\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAuthor Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\maketitle\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eYour introduction text here.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"overleaf-explains-that-every-latex-document-must-begin-with-a-documentclass-declaration-and-end-with-enddocument\" class=\"headerLink\"\u003e\n \u003ca href=\"#overleaf-explains-that-every-latex-document-must-begin-with-a-documentclass-declaration-and-end-with-enddocument\" class=\"header-mark\" aria-label=\"Direct link to section: Overleaf explains that every LaTeX document must begin with a \\documentclass{} declaration and end with \\end{document}.\"\u003e\u003c/a\u003eOverleaf explains that every LaTeX document \u003cstrong\u003emust begin\u003c/strong\u003e with a \u003ccode\u003e\\documentclass{}\u003c/code\u003e declaration and end with \u003ccode\u003e\\end{document}\u003c/code\u003e.\u003c/h2\u003e\u003ch2 id=\"preamble\" class=\"headerLink\"\u003e\n \u003ca href=\"#preamble\" class=\"header-mark\" aria-label=\"Direct link to section: Preamble\"\u003e\u003c/a\u003ePreamble\u003c/h2\u003e\u003cp\u003eThe preamble contains all initial settings before the document begins. It typically includes:\u003c/p\u003e\n\u003ch3 id=\"document-class\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-class\" class=\"header-mark\" aria-label=\"Direct link to section: Document Class\"\u003e\u003c/a\u003eDocument Class\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[12pt, a4paper]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eClass\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eDescription\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003earticle\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFor articles, short reports, or papers\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ereport\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFor long reports, final projects, or dissertations\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ebook\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFor books with chapters\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003eletter\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFor formal letters\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"some-standard-classes\" class=\"headerLink\"\u003e\n \u003ca href=\"#some-standard-classes\" class=\"header-mark\" aria-label=\"Direct link to section: The document class determines the type and general style of the document.\nSome standard classes:\"\u003e\u003c/a\u003eThe document class determines the type and general style of the document.\nSome standard classes:\u003c/h2\u003e\u003ch3 id=\"packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#packages\" class=\"header-mark\" aria-label=\"Direct link to section: Packages\"\u003e\u003c/a\u003ePackages\u003c/h3\u003e\u003cp\u003ePackages extend LaTeX\u0026rsquo;s capabilities.\nCommon examples:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[utf8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003einputenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% Set text encoding\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[T1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efontenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% Set font encoding\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% Insert images\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath, amssymb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e Mathematical symbols and formulas\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch3 id=\"document-metadata\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-metadata\" class=\"header-mark\" aria-label=\"Direct link to section: Document Metadata\"\u003e\u003c/a\u003eDocument Metadata\u003c/h3\u003e\u003cp\u003eMetadata contains information such as title, author, and date.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eLaTeX Document Example\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAuthor Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e or can be left empty with \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"this-metadata-is-used-when-the-maketitle-command-is-called-in-the-document-body\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-metadata-is-used-when-the-maketitle-command-is-called-in-the-document-body\" class=\"header-mark\" aria-label=\"Direct link to section: This metadata is used when the \\maketitle command is called in the document body.\"\u003e\u003c/a\u003eThis metadata is used when the \u003ccode\u003e\\maketitle\u003c/code\u003e command is called in the document body.\u003c/h2\u003e\u003ch2 id=\"document-body\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-body\" class=\"header-mark\" aria-label=\"Direct link to section: Document Body\"\u003e\u003c/a\u003eDocument Body\u003c/h2\u003e\u003cp\u003eThis section contains the main content between \u003ccode\u003e\\begin{document}\u003c/code\u003e and \u003ccode\u003e\\end{document}\u003c/code\u003e.\nExample basic structure:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\maketitle\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eOpening or introductory section of the topic.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eMethodology\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eExplaining research methods or steps.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\subsection\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSteps\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMore detailed explanation of the method.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eConclusion\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eFinal conclusion of the discussion.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"section-numbering\" class=\"headerLink\"\u003e\n \u003ca href=\"#section-numbering\" class=\"header-mark\" aria-label=\"Direct link to section: Section Numbering\"\u003e\u003c/a\u003eSection Numbering\u003c/h3\u003e\u003cp\u003eLaTeX automatically numbers sections (\u003ccode\u003e\\section\u003c/code\u003e, \u003ccode\u003e\\subsection\u003c/code\u003e, etc.).\nFor unnumbered sections, use an asterisk:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section*\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAcknowledgments\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch3 id=\"adding-images\" class=\"headerLink\"\u003e\n \u003ca href=\"#adding-images\" class=\"header-mark\" aria-label=\"Direct link to section: Adding Images\"\u003e\u003c/a\u003eAdding Images\u003c/h3\u003e\u003cp\u003eUse the \u003ccode\u003egraphicx\u003c/code\u003e package:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.5\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eExample Image\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\label\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efig:example\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch3 id=\"writing-formulas-and-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#writing-formulas-and-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Writing Formulas and Equations\"\u003e\u003c/a\u003eWriting Formulas and Equations\u003c/h3\u003e\u003cp\u003eUse the \u003ccode\u003eamsmath\u003c/code\u003e package:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eequation\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eE = mc\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eequation\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"additional-sections-optional\" class=\"headerLink\"\u003e\n \u003ca href=\"#additional-sections-optional\" class=\"header-mark\" aria-label=\"Direct link to section: Additional Sections (Optional)\"\u003e\u003c/a\u003eAdditional Sections (Optional)\u003c/h2\u003e\u003cp\u003eFor academic documents or books, LaTeX provides additional divisions such as:\u003c/p\u003e\n\u003ch3 id=\"frontmatter-mainmatter-and-backmatter\" class=\"headerLink\"\u003e\n \u003ca href=\"#frontmatter-mainmatter-and-backmatter\" class=\"header-mark\" aria-label=\"Direct link to section: Frontmatter, Mainmatter, and Backmatter\"\u003e\u003c/a\u003eFrontmatter, Mainmatter, and Backmatter\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\frontmatter\u003c/span\u003e \u003cspan class=\"c\"\u003e% For front matter (abstract, table of contents)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tableofcontents\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\mainmatter\u003c/span\u003e \u003cspan class=\"c\"\u003e% Main content (chapters, subchapters)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\chapter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\backmatter\u003c/span\u003e \u003cspan class=\"c\"\u003e% Back matter (bibliography, appendices)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\appendix\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\chapter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAppendix\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch3 id=\"bibliography\" class=\"headerLink\"\u003e\n \u003ca href=\"#bibliography\" class=\"header-mark\" aria-label=\"Direct link to section: Bibliography\"\u003e\u003c/a\u003eBibliography\u003c/h3\u003e\u003cp\u003eUse \u003ccode\u003ebiblatex\u003c/code\u003e for automatic bibliography:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebiblatex\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\addbibresource\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebibliography.bib\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\printbibliography\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"minimal-document-example\" class=\"headerLink\"\u003e\n \u003ca href=\"#minimal-document-example\" class=\"header-mark\" aria-label=\"Direct link to section: Minimal Document Example\"\u003e\u003c/a\u003eMinimal Document Example\u003c/h2\u003e\u003cp\u003eHere\u0026rsquo;s an example of the \u003cstrong\u003esimplest\u003c/strong\u003e valid document:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[utf8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003einputenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eHello World in LaTeX\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAuthor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\maketitle\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis is a minimal document example in LaTeX.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"tips-and-best-practices\" class=\"headerLink\"\u003e\n \u003ca href=\"#tips-and-best-practices\" class=\"header-mark\" aria-label=\"Direct link to section: Tips and Best Practices\"\u003e\u003c/a\u003eTips and Best Practices\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eUse \u003cstrong\u003eseparate folder structure\u003c/strong\u003e (\u003ccode\u003e/images\u003c/code\u003e, \u003ccode\u003e/chapters\u003c/code\u003e, \u003ccode\u003e/bib\u003c/code\u003e) to make large documents easier to manage.\u003c/li\u003e\n\u003cli\u003eSeparate each chapter into \u003ccode\u003e.tex\u003c/code\u003e files and call them with \u003ccode\u003e\\input{}\u003c/code\u003e or \u003ccode\u003e\\include{}\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eUse the \u003cstrong\u003eOverleaf Outline Panel\u003c/strong\u003e to quickly view document structure.\u003c/li\u003e\n\u003cli\u003eAvoid loading excessive unnecessary packages to maintain compilation performance.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"overleaf-provides-complete-guides-practical-examples-and-templates-that-make-it-easier-for-new-users-to-learn-and-write-professional-documents\" class=\"headerLink\"\u003e\n \u003ca href=\"#overleaf-provides-complete-guides-practical-examples-and-templates-that-make-it-easier-for-new-users-to-learn-and-write-professional-documents\" class=\"header-mark\" aria-label=\"Direct link to section: The basic structure of LaTeX documents is highly organized and modular. By understanding the division between preamble, body, and additional sections, writers can manage technical documents more efficiently. Overleaf provides complete guides, practical examples, and templates that make it easier for new users to learn and write professional documents.\"\u003e\u003c/a\u003eThe basic structure of LaTeX documents is highly organized and modular.\nBy understanding the division between \u003cstrong\u003epreamble\u003c/strong\u003e, \u003cstrong\u003ebody\u003c/strong\u003e, and \u003cstrong\u003eadditional sections\u003c/strong\u003e, writers can manage technical documents more efficiently.\nOverleaf provides complete guides, practical examples, and templates that make it easier for new users to learn and write professional documents.\u003c/h2\u003e\u003ch3 id=\"-references\" class=\"headerLink\"\u003e\n \u003ca href=\"#-references\" class=\"header-mark\" aria-label=\"Direct link to section: 📚 References\"\u003e\u003c/a\u003e📚 References\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\u003ca href=\"https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eLearn LaTeX in 30 Minutes – Overleaf\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.overleaf.com/learn/latex/Sections_and_chapters\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eSections and Chapters – Overleaf\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.overleaf.com/learn/latex/How_to_Write_a_Thesis_in_LaTeX_%28Part_1%29%3A_Basic_Structure\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eHow to Write a Thesis in LaTeX (Part 1): Basic Structure – Overleaf\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.overleaf.com/learn/latex/Inserting_Images\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eInserting Images – Overleaf\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.overleaf.com/learn/latex/Bibliography_management_with_biblatex\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eBibliography Management with BibLaTeX – Overleaf\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.overleaf.com/learn/how-to/Using_the_File_Outline_in_Overleaf\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eUsing the File Outline in Overleaf\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e", "language": "en" }, { "title" : "Installation Guide for MiKTeX, MacTeX, and TeX Live Across Operating Systems", "date_published" : "2025-10-08T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/latex-machine/", "url" : "https://www.aantriono.com/en/latex-machine/", "summary": "This article discusses how to install various LaTeX engines used for creating documents across different operating systems. To learn LaTeX in depth, visit LaTeX Learning Guide.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses how to install various LaTeX engines used for creating documents across different operating systems.\nTo learn LaTeX in depth, visit \u003ca href=\"https://www.aantriono.com/2022/07/buku-panduan-belajar-latex.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eLaTeX Learning Guide\u003c/a\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eWant to write your thesis, scientific report, or book with the appearance of an international academic publication?\u003c/strong\u003e\nThis guide will help you understand and install LaTeX on Windows, macOS, and Linux — from choosing the right distribution to solving common issues.\nPerfect for both beginners and advanced users who want to work efficiently and professionally.\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"a-brief-overview-of-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-brief-overview-of-latex\" class=\"header-mark\" aria-label=\"Direct link to section: A Brief Overview of LaTeX\"\u003e\u003c/a\u003eA Brief Overview of LaTeX\u003c/h2\u003e\u003cp\u003eLaTeX is a document preparation (typesetting) system designed to produce professional-quality text layouts.\nIt is widely used in academia, science, and publishing because it handles complex structures such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eMathematical formulas and scientific symbols\u003c/li\u003e\n\u003cli\u003eAutomatic citations and bibliographies\u003c/li\u003e\n\u003cli\u003eConsistent and clean document layouts\nTo run LaTeX, you need a \u003cem\u003eTeX distribution\u003c/em\u003e — a complete package containing all the components needed to write, compile, and format your document.\nThe three most commonly used distributions are \u003cstrong\u003eMiKTeX\u003c/strong\u003e (Windows), \u003cstrong\u003eMacTeX\u003c/strong\u003e (macOS), and \u003cstrong\u003eTeX Live\u003c/strong\u003e (cross-platform).\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"what-is-a-latex-distribution\" class=\"headerLink\"\u003e\n \u003ca href=\"#what-is-a-latex-distribution\" class=\"header-mark\" aria-label=\"Direct link to section: What Is a LaTeX Distribution?\"\u003e\u003c/a\u003eWhat Is a LaTeX Distribution?\u003c/h2\u003e\u003cp\u003eA LaTeX distribution is a collection of software that provides:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eTeX engines\u003c/strong\u003e such as pdfTeX, XeTeX, or LuaTeX\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePackage manager\u003c/strong\u003e for downloading and updating packages\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFont collections and document styles\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCompilers\u003c/strong\u003e to convert \u003ccode\u003e.tex\u003c/code\u003e files into \u003ccode\u003e.pdf\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBuilt-in editors\u003c/strong\u003e (optional) for writing LaTeX code\nIn short, a LaTeX distribution is the foundation you need to write and compile documents seamlessly.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"comparison-of-the-three-popular-distributions\" class=\"headerLink\"\u003e\n \u003ca href=\"#comparison-of-the-three-popular-distributions\" class=\"header-mark\" aria-label=\"Direct link to section: Comparison of the Three Popular Distributions\"\u003e\u003c/a\u003eComparison of the Three Popular Distributions\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eAspect\u003c/strong\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eMiKTeX\u003c/strong\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eMacTeX\u003c/strong\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eTeX Live\u003c/strong\u003e\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003ePlatform\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ePrimarily Windows (also Linux \u0026amp; macOS)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003emacOS only\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eWindows, macOS, Linux\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eInitial Size\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e±200 MB (basic version)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e±4 GB\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e±4 GB\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eInstallation Type\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eModular (on-demand)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFull installation\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFull or custom\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003ePackage Manager\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMiKTeX Console (GUI)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eTeX Live Utility\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003etlmgr\u003c/code\u003e (CLI)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eUpdate Frequency\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFrequent\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAnnual (with minor updates)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAnnual\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eEase of Use\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e★★★★★\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e★★★★☆\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e★★★☆☆\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eAuto-install Packages\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eYes (default)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNo\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCan be enabled\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eBuilt-in IDE\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eTeXworks\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eTeXShop\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eTeXworks\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eSize After Installation\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e1–4 GB\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e±7 GB\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e5–7 GB\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eLicense\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eOpen Source\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eOpen Source\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eOpen Source\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eUser Community\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLarge\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eMac-focused\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eVery large\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eRecommended For\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eWindows \u0026amp; beginners\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003emacOS users\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAdvanced users \u0026amp; Linux\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"installing-miktex-windows\" class=\"headerLink\"\u003e\n \u003ca href=\"#installing-miktex-windows\" class=\"header-mark\" aria-label=\"Direct link to section: Installing MiKTeX (Windows)\"\u003e\u003c/a\u003eInstalling MiKTeX (Windows)\u003c/h2\u003e\u003ch3 id=\"step-1-download-the-installer\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-1-download-the-installer\" class=\"header-mark\" aria-label=\"Direct link to section: Step 1: Download the Installer\"\u003e\u003c/a\u003eStep 1: Download the Installer\u003c/h3\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/latex-machine/miktex.webp\" title=\"\" data-thumbnail=\"/latex-machine/miktex.webp\" data-sub-html=\"\u003ch2\u003eMiKTeX Homepage\u003c/h2\u003e\"\u003e\u003cimg loading=\"lazy\" src='/latex-machine/miktex.webp' srcset=\"/latex-machine/miktex_hu_f298feb6ca219eed.webp 800w, /latex-machine/miktex_hu_26875edfc7efb30f.webp 1200w, /latex-machine/miktex_hu_322cd3a561880978.webp 1600w\" height=\"1562\" width=\"2400\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eMiKTeX Homepage\u003c/figcaption\u003e\n \u003c/figure\u003e\n\u003col\u003e\n\u003cli\u003eGo to \u003ca href=\"https://miktex.org/download\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://miktex.org/download\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eChoose the \u003cstrong\u003eWindows\u003c/strong\u003e version\u003c/li\u003e\n\u003cli\u003eDownload the installer (recommended: \u003cstrong\u003eMiKTeX-pdfTeX Installer\u003c/strong\u003e)\u003c/li\u003e\n\u003cli\u003eFile size is approximately 200 MB\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"step-2-run-the-installer\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-2-run-the-installer\" class=\"header-mark\" aria-label=\"Direct link to section: Step 2: Run the Installer\"\u003e\u003c/a\u003eStep 2: Run the Installer\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eDouble-click the installer file\u003c/li\u003e\n\u003cli\u003eChoose \u003cem\u003e“Install just for me”\u003c/em\u003e (no I rights needed)\u003c/li\u003e\n\u003cli\u003eFollow the installation wizard until completion\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"step-3-set-preferences\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-3-set-preferences\" class=\"header-mark\" aria-label=\"Direct link to section: Step 3: Set Preferences\"\u003e\u003c/a\u003eStep 3: Set Preferences\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eInstall missing packages on-the-fly\u003c/strong\u003e → select \u003cem\u003eYes\u003c/em\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePaper size\u003c/strong\u003e → choose \u003cem\u003eA4\u003c/em\u003e\u003c/li\u003e\n\u003cli\u003eClick \u003cem\u003eStart\u003c/em\u003e to begin installation\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"step-4-verify-installation\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-4-verify-installation\" class=\"header-mark\" aria-label=\"Direct link to section: Step 4: Verify Installation\"\u003e\u003c/a\u003eStep 4: Verify Installation\u003c/h3\u003e\u003cp\u003eOpen \u003cstrong\u003eCommand Prompt\u003c/strong\u003e and run:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003epdflatex --version\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf MiKTeX version information appears, installation was successful.\u003c/p\u003e\n\u003ch3 id=\"step-5-update-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-5-update-packages\" class=\"header-mark\" aria-label=\"Direct link to section: Step 5: Update Packages\"\u003e\u003c/a\u003eStep 5: Update Packages\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eOpen \u003cstrong\u003eMiKTeX Console\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eClick \u003cstrong\u003eUpdates → Check for updates → Update now\u003c/strong\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"installing-mactex-macos\" class=\"headerLink\"\u003e\n \u003ca href=\"#installing-mactex-macos\" class=\"header-mark\" aria-label=\"Direct link to section: Installing MacTeX (macOS)\"\u003e\u003c/a\u003eInstalling MacTeX (macOS)\u003c/h2\u003e\u003ch3 id=\"step-1-download-the-package\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-1-download-the-package\" class=\"header-mark\" aria-label=\"Direct link to section: Step 1: Download the Package\"\u003e\u003c/a\u003eStep 1: Download the Package\u003c/h3\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/latex-machine/mactex.webp\" title=\"\" data-thumbnail=\"/latex-machine/mactex.webp\" data-sub-html=\"\u003ch2\u003eMacTeX Homepage\u003c/h2\u003e\"\u003e\u003cimg loading=\"lazy\" src='/latex-machine/mactex.webp' srcset=\"/latex-machine/mactex_hu_affba9ddfb8d778f.webp 800w, /latex-machine/mactex_hu_238ee1063b4b9094.webp 1200w, /latex-machine/mactex_hu_4440496a938eb59f.webp 1600w\" height=\"1562\" width=\"2400\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eMacTeX Homepage\u003c/figcaption\u003e\n \u003c/figure\u003e\n\u003cul\u003e\n\u003cli\u003eVisit \u003ca href=\"https://www.tug.org/mactex/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.tug.org/mactex/\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eDownload \u003cstrong\u003eMacTeX.pkg\u003c/strong\u003e (~4 GB)\u003c/li\u003e\n\u003cli\u003eFor a lightweight version, choose \u003cstrong\u003eBasicTeX\u003c/strong\u003e (~100 MB)\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"step-2-installation-process\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-2-installation-process\" class=\"header-mark\" aria-label=\"Direct link to section: Step 2: Installation Process\"\u003e\u003c/a\u003eStep 2: Installation Process\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eDouble-click \u003ccode\u003eMacTeX.pkg\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eFollow the on-screen instructions\u003c/li\u003e\n\u003cli\u003eEnter your password if required\u003c/li\u003e\n\u003cli\u003eWait 10–20 minutes until the process finishes\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"step-3-verify-installation\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-3-verify-installation\" class=\"header-mark\" aria-label=\"Direct link to section: Step 3: Verify Installation\"\u003e\u003c/a\u003eStep 3: Verify Installation\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003epdflatex --version\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output will show the pdfTeX version from TeX Live.\u003c/p\u003e\n\u003ch3 id=\"step-4-update-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-4-update-packages\" class=\"header-mark\" aria-label=\"Direct link to section: Step 4: Update Packages\"\u003e\u003c/a\u003eStep 4: Update Packages\u003c/h3\u003e\u003cp\u003eUse \u003cstrong\u003eTeX Live Utility\u003c/strong\u003e, then select \u003cem\u003eUpdate All Packages\u003c/em\u003e.\u003c/p\u003e\n\u003ch3 id=\"step-5-launch-texshop\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-5-launch-texshop\" class=\"header-mark\" aria-label=\"Direct link to section: Step 5: Launch TeXShop\"\u003e\u003c/a\u003eStep 5: Launch TeXShop\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eInstalled automatically with MacTeX\u003c/li\u003e\n\u003cli\u003eLocation: \u003ccode\u003eApplications/TeX/TeXShop.app\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eA user-friendly editor for macOS users\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"installing-tex-live-on-linux\" class=\"headerLink\"\u003e\n \u003ca href=\"#installing-tex-live-on-linux\" class=\"header-mark\" aria-label=\"Direct link to section: Installing TeX Live on Linux\"\u003e\u003c/a\u003eInstalling TeX Live on Linux\u003c/h2\u003e\u003ch3 id=\"method-1-using-the-package-manager-recommended\" class=\"headerLink\"\u003e\n \u003ca href=\"#method-1-using-the-package-manager-recommended\" class=\"header-mark\" aria-label=\"Direct link to section: Method 1: Using the Package Manager (Recommended)\"\u003e\u003c/a\u003eMethod 1: Using the Package Manager (Recommended)\u003c/h3\u003e\u003ch2 id=\"ubuntudebian\" class=\"headerLink\"\u003e\n \u003ca href=\"#ubuntudebian\" class=\"header-mark\" aria-label=\"Direct link to section: Ubuntu/Debian\"\u003e\u003c/a\u003eUbuntu/Debian\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt update\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt install texlive-full\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"fedorarhel\" class=\"headerLink\"\u003e\n \u003ca href=\"#fedorarhel\" class=\"header-mark\" aria-label=\"Direct link to section: Fedora/RHEL\"\u003e\u003c/a\u003eFedora/RHEL\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo dnf install texlive-scheme-full\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"arch-linux\" class=\"headerLink\"\u003e\n \u003ca href=\"#arch-linux\" class=\"header-mark\" aria-label=\"Direct link to section: Arch Linux\"\u003e\u003c/a\u003eArch Linux\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo pacman -S texlive-most\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"method-2-manual-installation-advanced-users\" class=\"headerLink\"\u003e\n \u003ca href=\"#method-2-manual-installation-advanced-users\" class=\"header-mark\" aria-label=\"Direct link to section: Method 2: Manual Installation (Advanced Users)\"\u003e\u003c/a\u003eMethod 2: Manual Installation (Advanced Users)\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ewget https://mirror.ctan.org/systems/texlive/tlnet/install-tl-unx.tar.gz\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003etar -xzf install-tl-unx.tar.gz\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003ecd\u003c/span\u003e install-tl-*/\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo perl install-tl\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen, add PATH configuration to \u003ccode\u003e~/.bashrc\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003eexport\u003c/span\u003e \u003cspan class=\"nv\"\u003ePATH\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e/usr/local/texlive/2024/bin/x86_64-linux:\u003cspan class=\"se\"\u003e\\(\u003c/span\u003ePATH\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eReload the shell:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003esource\u003c/span\u003e ~/.bashrc\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"installing-a-latex-editor-optional-but-recommended\" class=\"headerLink\"\u003e\n \u003ca href=\"#installing-a-latex-editor-optional-but-recommended\" class=\"header-mark\" aria-label=\"Direct link to section: Installing a LaTeX Editor (Optional but Recommended)\"\u003e\u003c/a\u003eInstalling a LaTeX Editor (Optional but Recommended)\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eEditor\u003c/strong\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003ePlatform\u003c/strong\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eKey Features\u003c/strong\u003e\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eTeXstudio\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eWindows, macOS, Linux\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFull-featured, PDF integration, auto-completion\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eOverleaf\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eOnline\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eReal-time collaboration, no installation\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eVS Code + LaTeX Workshop\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCross-platform\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eModern, flexible, Git integration\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eTeXworks\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eBundled\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLightweight and beginner-friendly\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eLyX\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eWindows, macOS, Linux\u003c/td\u003e\n \u003ctd style=\"text-align: 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tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[utf8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003einputenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[english]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eLaTeX Test Document\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eYour Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\maketitle\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis is my first LaTeX document.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSample Formula\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ePythagorean theorem: \u003cspan class=\"k\"\u003e\\)\u003c/span\u003ea\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2 + b\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2 = c\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"nb\"\u003e^\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\infty\u003c/span\u003e\u003cspan class=\"nb\"\u003e e^{\u003c/span\u003e\u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"nb\"\u003ex^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e} dx \u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\sqrt\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\pi\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}{\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\]\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{document}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eCompile it via terminal:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003epdflatex test.tex\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"if-testpdf-is-generated-your-system-is-ready-to-use\" class=\"headerLink\"\u003e\n \u003ca href=\"#if-testpdf-is-generated-your-system-is-ready-to-use\" class=\"header-mark\" aria-label=\"Direct link to section: Or compile directly from your editor (e.g., press F5 in TeXstudio). If test.pdf is generated, your system is ready to use.\"\u003e\u003c/a\u003eOr compile directly from your editor (e.g., press \u003cstrong\u003eF5\u003c/strong\u003e in TeXstudio).\nIf \u003ccode\u003etest.pdf\u003c/code\u003e is generated, your system is ready to use.\u003c/h2\u003e\u003ch2 id=\"common-issues-and-solutions\" class=\"headerLink\"\u003e\n \u003ca href=\"#common-issues-and-solutions\" class=\"header-mark\" aria-label=\"Direct link to section: Common Issues and Solutions\"\u003e\u003c/a\u003eCommon Issues and Solutions\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eIssue\u003c/strong\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e\u003cstrong\u003eSolution\u003c/strong\u003e\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003ePackage not found\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eEnable \u003cem\u003eauto-install\u003c/em\u003e (MiKTeX) or run \u003ccode\u003etlmgr install \u0026lt;package-name\u0026gt;\u003c/code\u003e (TeX Live)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003epdflatex\u003c/code\u003e command not recognized\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eEnsure PATH is set correctly and restart the terminal\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eFont errors\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eRun \u003ccode\u003esudo fc-cache -fv\u003c/code\u003e on Linux or reinstall MiKTeX on Windows\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eCompilation failed\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCheck the \u003ccode\u003e.log\u003c/code\u003e file for details and ensure all packages are installed\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"tips--recommendations\" class=\"headerLink\"\u003e\n \u003ca href=\"#tips--recommendations\" class=\"header-mark\" aria-label=\"Direct link to section: Tips \u0026amp; Recommendations\"\u003e\u003c/a\u003eTips \u0026amp; Recommendations\u003c/h2\u003e\u003ch3 id=\"for-beginners\" class=\"headerLink\"\u003e\n \u003ca href=\"#for-beginners\" class=\"header-mark\" aria-label=\"Direct link to section: For Beginners:\"\u003e\u003c/a\u003eFor Beginners:\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eUse \u003cstrong\u003eMiKTeX + TeXstudio\u003c/strong\u003e (Windows)\u003c/li\u003e\n\u003cli\u003eUse \u003cstrong\u003eMacTeX + TeXShop\u003c/strong\u003e (macOS)\u003c/li\u003e\n\u003cli\u003eUse \u003cstrong\u003eTeX Live + TeXstudio\u003c/strong\u003e (Linux)\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"for-advanced-users\" class=\"headerLink\"\u003e\n \u003ca href=\"#for-advanced-users\" class=\"header-mark\" aria-label=\"Direct link to section: For Advanced Users:\"\u003e\u003c/a\u003eFor Advanced Users:\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eChoose \u003cstrong\u003eTeX Live\u003c/strong\u003e for cross-platform consistency\u003c/li\u003e\n\u003cli\u003eUse \u003cstrong\u003eVS Code + LaTeX Workshop\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eStore projects in Git for version control and collaboration\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"maintenance\" class=\"headerLink\"\u003e\n \u003ca href=\"#maintenance\" class=\"header-mark\" aria-label=\"Direct link to section: Maintenance:\"\u003e\u003c/a\u003eMaintenance:\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eUpdate packages regularly\u003c/li\u003e\n\u003cli\u003eClean up temporary files (\u003ccode\u003e.aux\u003c/code\u003e, \u003ccode\u003e.log\u003c/code\u003e)\u003c/li\u003e\n\u003cli\u003eBackup your templates and custom styles\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"additional-learning-resources\" class=\"headerLink\"\u003e\n \u003ca href=\"#additional-learning-resources\" class=\"header-mark\" aria-label=\"Direct link to section: Additional Learning Resources\"\u003e\u003c/a\u003eAdditional Learning Resources\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003ca href=\"https://www.latex-project.org/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eLaTeX Project\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://ctan.org/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eCTAN (Comprehensive TeX Archive Network)\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://en.wikibooks.org/wiki/LaTeX\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eLaTeX Wikibook\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.tug.org/texlive/doc.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeX Live Documentation\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://tex.stackexchange.com/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeX Stack Exchange\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eYour choice of LaTeX distribution depends on your needs and operating system:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eMiKTeX\u003c/strong\u003e → lightweight and beginner-friendly\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMacTeX\u003c/strong\u003e → stable and well-integrated for macOS users\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTeX Live\u003c/strong\u003e → versatile and ideal for all platforms\nOnce installed, you can start writing academic papers, journal articles, or professional books with precise and elegant typesetting.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eHappy writing with LaTeX — a powerful, clean, and elegant document system.\u003c/strong\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e", "language": "en" }, { "title" : "Getting to Know Ubuntu: A User-Friendly Linux Distribution for All Needs", "date_published" : "2025-10-06T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/closer-to-ubuntu/", "url" : "https://www.aantriono.com/en/closer-to-ubuntu/", "summary": "This article introduces Ubuntu, a popular, user-friendly Linux distribution designed for diverse use cases — from personal computing to professional development.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article introduces \u003cstrong\u003eUbuntu\u003c/strong\u003e, a popular, user-friendly Linux distribution designed for diverse use cases — from personal computing to professional development.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn a world dominated by paid operating systems, \u003cstrong\u003eUbuntu\u003c/strong\u003e emerges as a free, secure, and efficient alternative. As one of the most popular Linux distributions worldwide, it has proven itself a reliable and accessible choice for both beginners and IT professionals.\u003c/p\u003e\n\u003ch2 id=\"what-exactly-is-ubuntu\" class=\"headerLink\"\u003e\n \u003ca href=\"#what-exactly-is-ubuntu\" class=\"header-mark\" aria-label=\"Direct link to section: What Exactly Is Ubuntu?\"\u003e\u003c/a\u003eWhat Exactly Is Ubuntu?\u003c/h2\u003e\u003cp\u003eUbuntu is a \u003cstrong\u003eLinux-based operating system\u003c/strong\u003e developed by \u003cstrong\u003eCanonical Ltd.\u003c/strong\u003e, a company founded by South African entrepreneur \u003cstrong\u003eMark Shuttleworth\u003c/strong\u003e. The name “Ubuntu” originates from an African philosophy meaning \u003cem\u003e“humanity towards others”\u003c/em\u003e or \u003cem\u003e“I am because we all are,”\u003c/em\u003e reflecting the community spirit at the core of this project.\nBuilt upon \u003cstrong\u003eDebian\u003c/strong\u003e, one of the oldest and most stable Linux distributions, Ubuntu takes Debian’s robustness and repackages it with a user-friendly interface—making it ideal for newcomers to the Linux ecosystem.\u003c/p\u003e\n\u003ch2 id=\"key-features-of-ubuntu\" class=\"headerLink\"\u003e\n \u003ca href=\"#key-features-of-ubuntu\" class=\"header-mark\" aria-label=\"Direct link to section: Key Features of Ubuntu\"\u003e\u003c/a\u003eKey Features of Ubuntu\u003c/h2\u003e\u003ch3 id=\"open-source-and-free\" class=\"headerLink\"\u003e\n \u003ca href=\"#open-source-and-free\" class=\"header-mark\" aria-label=\"Direct link to section: Open Source and Free\"\u003e\u003c/a\u003eOpen Source and Free\u003c/h3\u003e\u003cp\u003eUnlike commercial operating systems, Ubuntu is completely free to download, use, and distribute. Its open-source nature allows anyone to view, modify, and improve the source code. It’s not just about cost—it’s about \u003cstrong\u003efreedom and transparency\u003c/strong\u003e in technology.\u003c/p\u003e\n\u003ch3 id=\"strong-security\" class=\"headerLink\"\u003e\n \u003ca href=\"#strong-security\" class=\"header-mark\" aria-label=\"Direct link to section: Strong Security\"\u003e\u003c/a\u003eStrong Security\u003c/h3\u003e\u003cp\u003eUbuntu is designed with \u003cstrong\u003esecurity\u003c/strong\u003e as a top priority. It includes a built-in firewall, data encryption, and automatic security updates. Thanks to Linux’s architecture, Ubuntu is naturally resistant to viruses and malware, making it a safe choice for all types of users.\u003c/p\u003e\n\u003ch3 id=\"stability-and-performance\" class=\"headerLink\"\u003e\n \u003ca href=\"#stability-and-performance\" class=\"header-mark\" aria-label=\"Direct link to section: Stability and Performance\"\u003e\u003c/a\u003eStability and Performance\u003c/h3\u003e\u003cp\u003eUbuntu offers \u003cstrong\u003eLong-Term Support (LTS)\u003c/strong\u003e versions, supported for \u003cstrong\u003efive years\u003c/strong\u003e with security updates. These releases ensure long-term stability—ideal for businesses and users who value reliability.\u003c/p\u003e\n\u003ch3 id=\"intuitive-user-interface\" class=\"headerLink\"\u003e\n \u003ca href=\"#intuitive-user-interface\" class=\"header-mark\" aria-label=\"Direct link to section: Intuitive User Interface\"\u003e\u003c/a\u003eIntuitive User Interface\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/closer-to-ubuntu/ubuntu.webp' srcset=\"/closer-to-ubuntu/ubuntu_hu_888fd50b0a7c6cc0.webp 800w, /closer-to-ubuntu/ubuntu_hu_fcbf754224326390.webp 1200w, /closer-to-ubuntu/ubuntu_hu_bd984a0cc4006866.webp 1600w\" alt=\"Ubuntu Desktop GNOME\" height=\"768\" width=\"1366\"\u003e\n\u003cem\u003eUbuntu 25.04 desktop running the modern GNOME environment\u003c/em\u003e\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eNote:\u003c/strong\u003e For the latest screenshots, visit \u003ca href=\"https://ubuntu.com/desktop\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eubuntu.com/desktop\u003c/a\u003e\nUbuntu uses the \u003cstrong\u003eGNOME\u003c/strong\u003e desktop environment by default, offering a modern and elegant experience. The interface is intuitive even for users transitioning from other operating systems. The left-hand dock provides quick access to favorite apps, while the Activities Overview allows fast searching of files, apps, and settings.\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"ubuntu-flavors-for-every-need\" class=\"headerLink\"\u003e\n \u003ca href=\"#ubuntu-flavors-for-every-need\" class=\"header-mark\" aria-label=\"Direct link to section: Ubuntu Flavors for Every Need\"\u003e\u003c/a\u003eUbuntu Flavors for Every Need\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/closer-to-ubuntu/flavour.webp' srcset=\"/closer-to-ubuntu/flavour_hu_f8df44548184f46a.webp 800w, /closer-to-ubuntu/flavour_hu_3390c41219405121.webp 1200w, /closer-to-ubuntu/flavour_hu_1b6c9a9f87fc237e.webp 1600w\" alt=\"Ubuntu Flavors\" height=\"720\" width=\"1280\"\u003e\n\u003cem\u003eVarious Ubuntu flavors designed for specific use cases\u003c/em\u003e\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eNote:\u003c/strong\u003e Explore all flavors at \u003ca href=\"https://ubuntu.com/desktop/flavours\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eubuntu.com/desktop/flavours\u003c/a\u003e\nUbuntu comes in several editions tailored to different needs:\u003c/p\u003e\n\u003c/blockquote\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eUbuntu Desktop\u003c/strong\u003e – For personal and office use, with a complete graphical interface.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eUbuntu Server\u003c/strong\u003e – Optimized for servers and cloud computing, without a GUI for efficiency.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eUbuntu Studio\u003c/strong\u003e – Geared toward content creators, including professional tools for audio, video, and graphics.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eKubuntu, Xubuntu, Lubuntu\u003c/strong\u003e – Variants using different desktop environments (KDE, Xfce, LXQt) for various performance and aesthetic preferences.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"advantages-of-using-ubuntu\" class=\"headerLink\"\u003e\n \u003ca href=\"#advantages-of-using-ubuntu\" class=\"header-mark\" aria-label=\"Direct link to section: Advantages of Using Ubuntu\"\u003e\u003c/a\u003eAdvantages of Using Ubuntu\u003c/h2\u003e\u003ch3 id=\"active-and-supportive-community\" class=\"headerLink\"\u003e\n \u003ca href=\"#active-and-supportive-community\" class=\"header-mark\" aria-label=\"Direct link to section: Active and Supportive Community\"\u003e\u003c/a\u003eActive and Supportive Community\u003c/h3\u003e\u003cp\u003eUbuntu’s \u003cstrong\u003ecommunity\u003c/strong\u003e is one of its greatest strengths. You’ll find active forums, detailed documentation, and countless tutorials in many languages. When an issue arises, someone in the community has likely faced and solved it.\u003c/p\u003e\n\u003ch3 id=\"wide-hardware-compatibility\" class=\"headerLink\"\u003e\n \u003ca href=\"#wide-hardware-compatibility\" class=\"header-mark\" aria-label=\"Direct link to section: Wide Hardware Compatibility\"\u003e\u003c/a\u003eWide Hardware Compatibility\u003c/h3\u003e\u003cp\u003eUbuntu supports a vast range of hardware—from old laptops to modern workstations. Most drivers are included out of the box, minimizing manual setup.\u003c/p\u003e\n\u003ch3 id=\"rich-application-ecosystem\" class=\"headerLink\"\u003e\n \u003ca href=\"#rich-application-ecosystem\" class=\"header-mark\" aria-label=\"Direct link to section: Rich Application Ecosystem\"\u003e\u003c/a\u003eRich Application Ecosystem\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/closer-to-ubuntu/software.webp' srcset=\"/closer-to-ubuntu/software_hu_41284a5550eb2e1c.webp 800w, /closer-to-ubuntu/software_hu_db7af10548b4236e.webp 1200w, /closer-to-ubuntu/software_hu_68226156e9f25383.webp 1600w\" alt=\"Ubuntu Software Center\" height=\"685\" width=\"1004\"\u003e\n\u003cem\u003eUbuntu Software Center offers thousands of free apps through an intuitive interface.\u003c/em\u003e\nThe \u003cstrong\u003eUbuntu Software Center\u003c/strong\u003e lets you easily install thousands of free applications—no command line required. From office suites like \u003cstrong\u003eLibreOffice\u003c/strong\u003e to browsers like \u003cstrong\u003eFirefox\u003c/strong\u003e and \u003cstrong\u003eChrome\u003c/strong\u003e, most tools are available with a few clicks.\u003c/p\u003e\n\u003ch3 id=\"ideal-for-software-development\" class=\"headerLink\"\u003e\n \u003ca href=\"#ideal-for-software-development\" class=\"header-mark\" aria-label=\"Direct link to section: Ideal for Software Development\"\u003e\u003c/a\u003eIdeal for Software Development\u003c/h3\u003e\u003cp\u003eUbuntu is a favorite among developers. It supports almost all major programming languages and development tools, making it perfect for \u003cstrong\u003eweb, mobile, and desktop\u003c/strong\u003e development.\u003c/p\u003e\n\u003ch2 id=\"release-cycle-and-support\" class=\"headerLink\"\u003e\n \u003ca href=\"#release-cycle-and-support\" class=\"header-mark\" aria-label=\"Direct link to section: Release Cycle and Support\"\u003e\u003c/a\u003eRelease Cycle and Support\u003c/h2\u003e\u003cp\u003eUbuntu releases a new version every \u003cstrong\u003eApril and October\u003c/strong\u003e, named based on the year and month—e.g., \u003cem\u003eUbuntu 25.04\u003c/em\u003e (April 2025). Each version has a unique codename, such as \u003cem\u003e“Plucky Puffin.”\u003c/em\u003e\nUsers can choose between:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eRegular releases\u003c/strong\u003e (9-month support)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLTS releases\u003c/strong\u003e (5-year support)\nFor production environments, \u003cstrong\u003eLTS\u003c/strong\u003e versions are highly recommended.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"who-should-use-ubuntu\" class=\"headerLink\"\u003e\n \u003ca href=\"#who-should-use-ubuntu\" class=\"header-mark\" aria-label=\"Direct link to section: Who Should Use Ubuntu?\"\u003e\u003c/a\u003eWho Should Use Ubuntu?\u003c/h2\u003e\u003cp\u003eUbuntu is suitable for:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eLinux beginners\u003c/strong\u003e exploring an easy-to-learn alternative OS\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eIT professionals\u003c/strong\u003e seeking a stable development or server platform\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eEducational institutions\u003c/strong\u003e wanting a cost-effective, reliable solution\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBusinesses\u003c/strong\u003e reducing licensing costs without compromising security\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eContent creators\u003c/strong\u003e using multimedia tools without expensive software\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"ubuntu-installation-guide\" class=\"headerLink\"\u003e\n \u003ca href=\"#ubuntu-installation-guide\" class=\"header-mark\" aria-label=\"Direct link to section: Ubuntu Installation Guide\"\u003e\u003c/a\u003eUbuntu Installation Guide\u003c/h2\u003e\u003cp\u003eInstalling Ubuntu is straightforward—even for beginners.\u003c/p\u003e\n\u003ch3 id=\"preparation-before-installation\" class=\"headerLink\"\u003e\n \u003ca href=\"#preparation-before-installation\" class=\"header-mark\" aria-label=\"Direct link to section: Preparation Before Installation\"\u003e\u003c/a\u003ePreparation Before Installation\u003c/h3\u003e\u003ch2 id=\"1-backup-important-data\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-backup-important-data\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Backup Important Data\"\u003e\u003c/a\u003e1. Backup Important Data\u003c/h2\u003e\u003cp\u003eAlways back up essential data before installation, whether to external media or the cloud.\u003c/p\u003e\n\u003ch2 id=\"2-minimum-system-requirements\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-minimum-system-requirements\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Minimum System Requirements\"\u003e\u003c/a\u003e2. Minimum System Requirements\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e2 GHz dual-core processor or higher\u003c/li\u003e\n\u003cli\u003e4 GB RAM (8 GB recommended)\u003c/li\u003e\n\u003cli\u003e25 GB disk space minimum\u003c/li\u003e\n\u003cli\u003eUSB or DVD drive\u003c/li\u003e\n\u003cli\u003eInternet connection (recommended)\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"3-download-ubuntu-iso-file\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-download-ubuntu-iso-file\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Download Ubuntu ISO File\"\u003e\u003c/a\u003e3. Download Ubuntu ISO File\u003c/h2\u003e\u003cp\u003eVisit \u003ca href=\"https://ubuntu.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eubuntu.com\u003c/a\u003e and download the desired ISO—preferably the latest LTS version.\u003c/p\u003e\n\u003ch3 id=\"creating-a-bootable-installation-media\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-a-bootable-installation-media\" class=\"header-mark\" aria-label=\"Direct link to section: Creating a Bootable Installation Media\"\u003e\u003c/a\u003eCreating a Bootable Installation Media\u003c/h3\u003e\u003ch2 id=\"for-usb-drive\" class=\"headerLink\"\u003e\n \u003ca href=\"#for-usb-drive\" class=\"header-mark\" aria-label=\"Direct link to section: For USB Drive\"\u003e\u003c/a\u003eFor USB Drive\u003c/h2\u003e\u003cp\u003eUse \u003cstrong\u003eRufus\u003c/strong\u003e (Windows), \u003cstrong\u003ebalenaEtcher\u003c/strong\u003e (cross-platform), or \u003cstrong\u003eStartup Disk Creator\u003c/strong\u003e (Linux) to create a bootable USB.\nSteps using balenaEtcher:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eOpen balenaEtcher\u003c/li\u003e\n\u003cli\u003eSelect the downloaded Ubuntu ISO\u003c/li\u003e\n\u003cli\u003eChoose your USB drive\u003c/li\u003e\n\u003cli\u003eClick “Flash” and wait for completion\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"step-by-step-installation\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-installation\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step Installation\"\u003e\u003c/a\u003eStep-by-Step Installation\u003c/h3\u003e\u003ch2 id=\"step-1-boot-from-installation-media\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-1-boot-from-installation-media\" class=\"header-mark\" aria-label=\"Direct link to section: Step 1: Boot from Installation Media\"\u003e\u003c/a\u003eStep 1: Boot from Installation Media\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eRestart your PC and enter BIOS/UEFI (press F2, F12, Del, or Esc)\u003c/li\u003e\n\u003cli\u003eSet USB/DVD as first boot priority\u003c/li\u003e\n\u003cli\u003eSave and reboot\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"step-2-start-ubuntu-installer\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-2-start-ubuntu-installer\" class=\"header-mark\" aria-label=\"Direct link to section: Step 2: Start Ubuntu Installer\"\u003e\u003c/a\u003eStep 2: Start Ubuntu Installer\u003c/h2\u003e\u003cp\u003eChoose:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cem\u003e“Try Ubuntu”\u003c/em\u003e — try it live without installing\u003c/li\u003e\n\u003cli\u003e\u003cem\u003e“Install Ubuntu”\u003c/em\u003e — begin installation\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/closer-to-ubuntu/instalasi.webp' srcset=\"/closer-to-ubuntu/instalasi_hu_6a042de0eeafc2b3.webp 800w, /closer-to-ubuntu/instalasi_hu_cf10b509a3acf42d.webp 1200w, /closer-to-ubuntu/instalasi_hu_794696639aaea4e6.webp 1600w\" alt=\"Ubuntu Partitioning Options\" height=\"389\" width=\"711\"\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"step-37\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-37\" class=\"header-mark\" aria-label=\"Direct link to section: Step 3–7\"\u003e\u003c/a\u003eStep 3–7\u003c/h2\u003e\u003cp\u003eFollow prompts for:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eLanguage \u0026amp; keyboard layout\u003c/li\u003e\n\u003cli\u003eInternet connection\u003c/li\u003e\n\u003cli\u003eInstallation type (erase disk, dual-boot, or manual partition)\u003c/li\u003e\n\u003cli\u003eTimezone selection\u003c/li\u003e\n\u003cli\u003eUser account creation\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"step-8-installation-process\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-8-installation-process\" class=\"header-mark\" aria-label=\"Direct link to section: Step 8: Installation Process\"\u003e\u003c/a\u003eStep 8: Installation Process\u003c/h2\u003e\u003cp\u003eUbuntu will copy files and install. This takes ~15–30 minutes.\u003c/p\u003e\n\u003ch2 id=\"step-9-restart\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-9-restart\" class=\"header-mark\" aria-label=\"Direct link to section: Step 9: Restart\"\u003e\u003c/a\u003eStep 9: Restart\u003c/h2\u003e\u003cp\u003eOnce complete, remove the USB and reboot into Ubuntu.\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003ca href=\"https://www.linuxbuzz.com/ubuntu-24-04-desktop-installation-guide\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eFull Ubuntu Installation Guide\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\n\u003ch3 id=\"post-installation-setup\" class=\"headerLink\"\u003e\n \u003ca href=\"#post-installation-setup\" class=\"header-mark\" aria-label=\"Direct link to section: Post-Installation Setup\"\u003e\u003c/a\u003ePost-Installation Setup\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://i.imgur.com/ubuntu-welcome-placeholder.png' alt=\"Ubuntu First Boot\" \u003e\n\u003cem\u003eUbuntu welcome screen after first boot.\u003c/em\u003e\u003c/p\u003e\n\u003ch2 id=\"1-update-the-system\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-update-the-system\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Update the System\"\u003e\u003c/a\u003e1. Update the System\u003c/h2\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt update\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esudo apt upgrade\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"2-install-additional-apps\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-install-additional-apps\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Install Additional Apps\"\u003e\u003c/a\u003e2. Install Additional Apps\u003c/h2\u003e\u003cp\u003eUse Ubuntu Software to install:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eChrome / Firefox\u003c/li\u003e\n\u003cli\u003eVLC Media Player\u003c/li\u003e\n\u003cli\u003eGIMP\u003c/li\u003e\n\u003cli\u003eLibreOffice\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"3-configure-drivers\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-configure-drivers\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Configure Drivers\"\u003e\u003c/a\u003e3. Configure Drivers\u003c/h2\u003e\u003cp\u003eGo to \u003cem\u003eSoftware \u0026amp; Updates → Additional Drivers\u003c/em\u003e for proprietary graphics drivers if needed.\u003c/p\u003e\n\u003ch2 id=\"4-personalize-the-desktop\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-personalize-the-desktop\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Personalize the Desktop\"\u003e\u003c/a\u003e4. Personalize the Desktop\u003c/h2\u003e\u003cp\u003eSettings → Appearance allows theme, wallpaper, and layout customization.\u003c/p\u003e\n\u003ch3 id=\"dual-boot-tips\" class=\"headerLink\"\u003e\n \u003ca href=\"#dual-boot-tips\" class=\"header-mark\" aria-label=\"Direct link to section: Dual-Boot Tips\"\u003e\u003c/a\u003eDual-Boot Tips\u003c/h3\u003e\u003cp\u003eIf installing Ubuntu alongside Windows:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eInstall \u003cstrong\u003eWindows first\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eDisable \u003cstrong\u003eFast Startup\u003c/strong\u003e in Windows\u003c/li\u003e\n\u003cli\u003eDisable \u003cstrong\u003eSecure Boot\u003c/strong\u003e if boot issues occur\u003c/li\u003e\n\u003cli\u003eUbuntu will automatically detect Windows and configure \u003cstrong\u003eGRUB\u003c/strong\u003e for dual-boot.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"for-anyone-looking-to-move-away-from-proprietary-systems-or-explore-linux-ubuntu-offers-the-perfect-entry-pointbalancing-ease-of-use-security-stability-and-freedom\" class=\"headerLink\"\u003e\n \u003ca href=\"#for-anyone-looking-to-move-away-from-proprietary-systems-or-explore-linux-ubuntu-offers-the-perfect-entry-pointbalancing-ease-of-use-security-stability-and-freedom\" class=\"header-mark\" aria-label=\"Direct link to section: Ubuntu is a comprehensive operating system combining powerful features, high performance, and user-friendly design. With strong community support, regular updates, and open-source principles, it remains a top choice for millions worldwide. For anyone looking to move away from proprietary systems or explore Linux, Ubuntu offers the perfect entry point—balancing ease of use, security, stability, and freedom.\"\u003e\u003c/a\u003eUbuntu is a comprehensive operating system combining \u003cstrong\u003epowerful features, high performance, and user-friendly design\u003c/strong\u003e. With strong community support, regular updates, and open-source principles, it remains a top choice for millions worldwide.\nFor anyone looking to move away from proprietary systems or explore Linux, Ubuntu offers the \u003cstrong\u003eperfect entry point\u003c/strong\u003e—balancing ease of use, security, stability, and freedom.\u003c/h2\u003e\u003cp\u003e\u003cem\u003eInterested in trying Ubuntu? Visit \u003ca href=\"https://ubuntu.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eubuntu.com\u003c/a\u003e to download the latest version and start your open-source journey today.\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "History of TeX and LaTeX: Evolution of Digital Document Typesetting Systems", "date_published" : "2025-10-06T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/history-of-tex-and-latex/", "url" : "https://www.aantriono.com/en/history-of-tex-and-latex/", "summary": "This article discusses the history of TeX and LaTeX, document typesetting systems by Knuth and Lamport, from their inception in 1978 to modern academic standards, focusing on development, features, and future prospects to date.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the history of TeX and LaTeX, document typesetting systems by Knuth and Lamport, from their inception in 1978 to modern academic standards, focusing on development, features, and future prospects to date.\u003c/p\u003e\n\u003chr\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eTo learn LaTeX more deeply, please visit \u003ca href=\"https://www.aantriono.com/2022/07/buku-panduan-belajar-latex.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eLaTeX Learning Guide\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003cblockquote\u003e\n \u003cp\u003eTeX and LaTeX are two document typesetting tools that have revolutionized the way technical and scientific documents are composed, especially in the fields of mathematics, physics, and computer science. This article will comprehensively discuss the complete, valid, and comprehensive history of TeX and LaTeX development, from their origins to their impact in the modern era.\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"the-beginning-background-of-tex\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-beginning-background-of-tex\" class=\"header-mark\" aria-label=\"Direct link to section: The Beginning: Background of TeX\"\u003e\u003c/a\u003eThe Beginning: Background of TeX\u003c/h2\u003e\u003cp\u003eTeX was created by Donald E. Knuth, a renowned computer scientist and professor emeritus at Stanford University. The idea to develop TeX emerged in the late 1970s when Knuth was dissatisfied with the print quality of his second book, \u003cem\u003eThe Art of Computer Programming\u003c/em\u003e, Volume 2. At that time, the publishing industry was transitioning from traditional typesetting technology (with metal) to digital photocomposition systems, which often produced low-quality output, especially for mathematical formulas.\nIn 1977, Knuth began an ambitious project to create a digital typesetting system that could produce high-quality documents, especially for texts containing complex mathematical notation. His goal was to create a tool that was not only aesthetic but also flexible and could be used by technical writers without requiring deep graphic design expertise. This project resulted in TeX, a typesetting system designed to produce documents with high typographic precision.\u003c/p\u003e\n\u003ch2 id=\"development-of-tex\" class=\"headerLink\"\u003e\n \u003ca href=\"#development-of-tex\" class=\"header-mark\" aria-label=\"Direct link to section: Development of TeX\"\u003e\u003c/a\u003eDevelopment of TeX\u003c/h2\u003e\u003ch3 id=\"first-release-1978\" class=\"headerLink\"\u003e\n \u003ca href=\"#first-release-1978\" class=\"header-mark\" aria-label=\"Direct link to section: First Release (1978)\"\u003e\u003c/a\u003eFirst Release (1978)\u003c/h3\u003e\u003cp\u003eKnuth released the initial version of TeX in 1978. The system was written in the SAIL programming language and ran on mainframe computers at Stanford. TeX was designed to provide high-level control over document layout, including spacing, fonts, and placement of elements such as mathematical formulas. One of TeX\u0026rsquo;s major innovations was the line-breaking algorithm, which ensured that text was arranged with optimal aesthetics, avoiding uneven spacing between words or overly dense lines.\u003c/p\u003e\n\u003ch3 id=\"tex82-1982\" class=\"headerLink\"\u003e\n \u003ca href=\"#tex82-1982\" class=\"header-mark\" aria-label=\"Direct link to section: TeX82 (1982)\"\u003e\u003c/a\u003eTeX82 (1982)\u003c/h3\u003e\u003cp\u003eThe early version of TeX had limitations, including dependence on specific hardware. To address this, Knuth rewrote TeX in 1982 using the Pascal programming language and introduced the Metafont-based font system, which allowed for the creation of scalable digital fonts. This version, known as TeX82, became the standard used to this day. Knuth also ensured that TeX was open-source, allowing the community to develop and modify the system.\u003c/p\u003e\n\u003ch3 id=\"main-features-of-tex\" class=\"headerLink\"\u003e\n \u003ca href=\"#main-features-of-tex\" class=\"header-mark\" aria-label=\"Direct link to section: Main Features of TeX\"\u003e\u003c/a\u003eMain Features of TeX\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eMathematical Typesetting:\u003c/strong\u003e TeX was designed to handle mathematical notation exceptionally well, allowing the writing of complex formulas with relatively simple syntax.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMetafont:\u003c/strong\u003e This system enables the creation of high-quality digital fonts that can be customized to document needs.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePortability:\u003c/strong\u003e TeX was designed to run on various platforms, making it a flexible tool for different computer systems.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eStability:\u003c/strong\u003e Knuth committed to maintaining TeX compatibility, so documents created with TeX can still be processed correctly even after decades.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"publication-and-adoption\" class=\"headerLink\"\u003e\n \u003ca href=\"#publication-and-adoption\" class=\"header-mark\" aria-label=\"Direct link to section: Publication and Adoption\"\u003e\u003c/a\u003ePublication and Adoption\u003c/h3\u003e\u003cp\u003eAfter the release of TeX82, the American Mathematical Society (AMS) adopted TeX as the standard typesetting tool for mathematical journals and publications. The scientific community, especially in mathematics, physics, and computer science, began using it widely due to its ability to produce beautiful and consistent documents.\u003c/p\u003e\n\u003ch2 id=\"birth-of-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#birth-of-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Birth of LaTeX\"\u003e\u003c/a\u003eBirth of LaTeX\u003c/h2\u003e\u003cp\u003eAlthough TeX was very powerful, its complex syntax and orientation toward low-level details made it difficult to use for non-technical users. To address this problem, Leslie Lamport, a computer scientist at SRI International, developed LaTeX in the early 1980 s.\u003c/p\u003e\n\u003ch3 id=\"latex-209-1985\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-209-1985\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX 2.09 (1985)\"\u003e\u003c/a\u003eLaTeX 2.09 (1985)\u003c/h3\u003e\u003cp\u003eLaTeX, which stands for \u0026ldquo;Lamport TeX,\u0026rdquo; is a set of TeX-based macros that simplify the document writing process. LaTeX allows users to focus on document content, such as structure (chapters, sections, subsections) and format (titles, table of contents, references), without needing to manually adjust typographic details. The first version of LaTeX, released in 1985, quickly became popular among academics due to its ease of use.\u003c/p\u003e\n\u003ch3 id=\"main-features-of-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#main-features-of-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Main Features of LaTeX\"\u003e\u003c/a\u003eMain Features of LaTeX\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eDocument Structure:\u003c/strong\u003e LaTeX introduced the concept of document classes (such as articles, books, or reports) and high-level commands for organizing documents with clear structure.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eReference Management:\u003c/strong\u003e LaTeX supports tools like BibTeX for automatically managing citations and bibliographies.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eExtensibility:\u003c/strong\u003e Users can create additional macros and packages to customize LaTeX according to their needs.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eVisual Consistency:\u003c/strong\u003e LaTeX ensures that documents have a consistent and professional layout without requiring design expertise.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"latex2e-1994\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex2e-1994\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX2e (1994)\"\u003e\u003c/a\u003eLaTeX2e (1994)\u003c/h3\u003e\u003cp\u003eIn 1994, a team led by Frank Mittelbach released LaTeX2e, a major update that improved LaTeX\u0026rsquo;s modularity and compatibility. LaTeX2e introduced a package system that allows users to add new functionality without modifying the core system. This version remains the standard to this day, with periodic updates to support new features such as Unicode fonts and integration with modern technology.\u003c/p\u003e\n\u003ch2 id=\"evolution-and-impact\" class=\"headerLink\"\u003e\n \u003ca href=\"#evolution-and-impact\" class=\"header-mark\" aria-label=\"Direct link to section: Evolution and Impact\"\u003e\u003c/a\u003eEvolution and Impact\u003c/h2\u003e\u003ch3 id=\"global-adoption\" class=\"headerLink\"\u003e\n \u003ca href=\"#global-adoption\" class=\"header-mark\" aria-label=\"Direct link to section: Global Adoption\"\u003e\u003c/a\u003eGlobal Adoption\u003c/h3\u003e\u003cp\u003eTeX and LaTeX have become the de facto standard for scientific document typesetting. Academic journals, book publishers, and educational institutions worldwide use LaTeX to produce high-quality documents. The open-source community has also produced thousands of additional packages, such as AMS-LaTeX for advanced mathematics, Beamer for presentations, and TikZ for graphics.\u003c/p\u003e\n\u003ch3 id=\"community-and-support\" class=\"headerLink\"\u003e\n \u003ca href=\"#community-and-support\" class=\"header-mark\" aria-label=\"Direct link to section: Community and Support\"\u003e\u003c/a\u003eCommunity and Support\u003c/h3\u003e\u003cp\u003eThe TeX and LaTeX community is very active, with organizations like TeX Users Group (TUG) founded in 1980 to support users and developers. Platforms like CTAN (Comprehensive TeX Archive Network) provide a centralized repository for software, packages, and documentation related to TeX and LaTeX.\u003c/p\u003e\n\u003ch3 id=\"modern-developments\" class=\"headerLink\"\u003e\n \u003ca href=\"#modern-developments\" class=\"header-mark\" aria-label=\"Direct link to section: Modern Developments\"\u003e\u003c/a\u003eModern Developments\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eXeTeX and LuaTeX:\u003c/strong\u003e To address font limitations and multilingual support, variants like XeTeX (2004) and LuaTeX (2007) were developed. XeTeX supports OpenType fonts and Unicode, while LuaTeX integrates the Lua programming language for greater flexibility.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eModern Editors and Tools:\u003c/strong\u003e Editors like TeXShop, Overleaf, and TeXworks have simplified LaTeX usage, especially with web-based interfaces like Overleaf that enable real-time collaboration.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eIntegration with New Technologies:\u003c/strong\u003e LaTeX is now used not only for printed documents but also for producing interactive PDFs, websites, and even mobile applications.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"impact-on-academia-and-industry\" class=\"headerLink\"\u003e\n \u003ca href=\"#impact-on-academia-and-industry\" class=\"header-mark\" aria-label=\"Direct link to section: Impact on Academia and Industry\"\u003e\u003c/a\u003eImpact on Academia and Industry\u003c/h3\u003e\u003cp\u003eLaTeX has become an essential tool in academia due to its ability to produce consistent and easily updatable documents. Many scientific conferences and journals require manuscripts to be composed in LaTeX to ensure quality and compatibility. Beyond academia, LaTeX is used in the publishing industry, technical report writing, and even creating resumes or legal documents.\u003c/p\u003e\n\u003ch2 id=\"challenges-and-criticism\" class=\"headerLink\"\u003e\n \u003ca href=\"#challenges-and-criticism\" class=\"header-mark\" aria-label=\"Direct link to section: Challenges and Criticism\"\u003e\u003c/a\u003eChallenges and Criticism\u003c/h2\u003e\u003cp\u003eAlthough TeX and LaTeX are very powerful, they are not without criticism:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eLearning Curve:\u003c/strong\u003e LaTeX syntax can feel daunting for beginners, especially for those accustomed to word processing software like Microsoft Word.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDebugging Complexity:\u003c/strong\u003e Errors in LaTeX code often produce error messages that are difficult to understand.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eVisual Limitations:\u003c/strong\u003e LaTeX is not designed for advanced graphic design, so users often need additional tools like Adobe Illustrator for complex graphics.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"future-of-tex-and-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#future-of-tex-and-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Future of TeX and LaTeX\"\u003e\u003c/a\u003eFuture of TeX and LaTeX\u003c/h2\u003e\u003cp\u003eAlthough document typesetting technology continues to evolve, TeX and LaTeX remain relevant due to their strong community and adaptability. Projects like LaTeX3, which have been in development since the 1990s, aim to update LaTeX\u0026rsquo;s core with a more modern and modular architecture. Additionally, integration with web-based technologies and artificial intelligence (such as automated writing tools) opens new opportunities for LaTeX use in the future.\u003c/p\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eTeX and LaTeX are important milestones in the history of digital document typesetting. Starting from Donald Knuth\u0026rsquo;s vision to create a high-quality typesetting system, TeX has evolved into a widely used tool around the world. LaTeX, with its more user-friendly approach, expanded TeX\u0026rsquo;s reach to a broader community. With active open-source community support and adaptation to modern technology, TeX and LaTeX will continue to be important tools for technical and scientific writers in the future.\u003c/p\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eKnuth, D. E. (1999). \u003cem\u003eThe TeXbook\u003c/em\u003e. Addison-Wesley.\u003c/li\u003e\n\u003cli\u003eLamport, L. (1994). \u003cem\u003eLaTeX: A Document Preparation System\u003c/em\u003e. Addison-Wesley.\u003c/li\u003e\n\u003cli\u003eOfficial TeX Users Group (TUG) website: \u003ca href=\"https://www.tug.org\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.tug.org\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eComprehensive TeX Archive Network (CTAN): \u003ca href=\"https://www.ctan.org\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.ctan.org\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e", "language": "en" }, { "title" : "Why Choose Linux", "date_published" : "2025-10-03T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/why-choose-linux/", "url" : "https://www.aantriono.com/en/why-choose-linux/", "summary": "This article discusses the reasons for migrating from Windows to Linux.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the reasons for migrating from Windows to Linux.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIt’s been 3 years since I started using Linux as my primary operating system for everyday work. It all began from the realization of the risks and ethical concerns of using \u003cstrong\u003epirated\u003c/strong\u003e or \u003cstrong\u003ecracked\u003c/strong\u003e software on Windows.\nThrough learning, both physically and digitally, and interacting with various Linux communities across the internet and Telegram, I was motivated to migrate from Windows to Linux. That’s when I decided to take Linux seriously.\nUbuntu was the first Linux distribution I tried. There was a \u003cstrong\u003edifferent feeling\u003c/strong\u003e seeing a desktop that looked unfamiliar compared to Windows. My curiosity grew instantly — many questions appeared that needed answers. Exploring deeper, I found \u003cstrong\u003eLibreOffice\u003c/strong\u003e as the default office suite, including Writer, Calc, Impress, Draw, Math, and Base.\nWas it difficult to use Linux applications at first? Of course — anything new requires \u003cstrong\u003eadaptation and patience\u003c/strong\u003e. 😄\nAs a new Linux user, I experienced what’s called \u003cstrong\u003e“distro hopping.”\u003c/strong\u003e This term refers to the habit of frequently switching between Linux distributions in a short period. Some of the distros I’ve tried include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eDebian\u003c/li\u003e\n\u003cli\u003eUbuntu\u003c/li\u003e\n\u003cli\u003eMX Linux\u003c/li\u003e\n\u003cli\u003eAntix\u003c/li\u003e\n\u003cli\u003eDeepin\u003c/li\u003e\n\u003cli\u003eArch\u003c/li\u003e\n\u003cli\u003eEndeavour\u003c/li\u003e\n\u003cli\u003eMabox\u003c/li\u003e\n\u003cli\u003eLangit Ketujuh\u003c/li\u003e\n\u003cli\u003eGaruda\u003c/li\u003e\n\u003cli\u003eFedora\nI also had a chance to try Unix-based systems such as:\u003c/li\u003e\n\u003cli\u003eFreeBSD\u003c/li\u003e\n\u003cli\u003eGhostBSD\nEven though I’ve switched between many distros, my Linux skill level… well, still about the same 😅\nHere’s a look at the desktop from the Linux distro I currently use. Guess which one?\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/why-choose-linux/manjaro.webp' srcset=\"/why-choose-linux/manjaro_hu_d70d85c6316a3f89.webp 800w, /why-choose-linux/manjaro_hu_afac81947809f112.webp 1200w, /why-choose-linux/manjaro_hu_520066486923afcf.webp 1600w\" alt=\"Manjaro\" height=\"1080\" width=\"1920\"\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"its-free\" class=\"headerLink\"\u003e\n \u003ca href=\"#its-free\" class=\"header-mark\" aria-label=\"Direct link to section: It’s Free\"\u003e\u003c/a\u003eIt’s Free\u003c/h2\u003e\u003ch2 id=\"unlike-windows-linux-doesnt-bother-you-with-activate-windows-messages-or-require-pirated-patches-that-could-contain-malware\" class=\"headerLink\"\u003e\n \u003ca href=\"#unlike-windows-linux-doesnt-bother-you-with-activate-windows-messages-or-require-pirated-patches-that-could-contain-malware\" class=\"header-mark\" aria-label=\"Direct link to section: Linux is completely free. You don’t need to pay for any license, except for enterprise editions like Red Hat Enterprise Linux (RHEL) or SUSE Linux Enterprise. Unlike Windows, Linux doesn’t bother you with “Activate Windows” messages or require pirated patches that could contain malware.\"\u003e\u003c/a\u003eLinux is completely \u003cstrong\u003efree\u003c/strong\u003e. You don’t need to pay for any license, except for enterprise editions like \u003cstrong\u003eRed Hat Enterprise Linux (RHEL)\u003c/strong\u003e or \u003cstrong\u003eSUSE Linux Enterprise\u003c/strong\u003e.\nUnlike Windows, Linux doesn’t bother you with “Activate Windows” messages or require pirated patches that could contain malware.\u003c/h2\u003e\u003ch2 id=\"more-secure\" class=\"headerLink\"\u003e\n \u003ca href=\"#more-secure\" class=\"header-mark\" aria-label=\"Direct link to section: More Secure\"\u003e\u003c/a\u003eMore Secure\u003c/h2\u003e\u003ch2 id=\"linuxs-package-management-repository-based-software-system-and-permission-model-all-contribute-to-its-strong-security-you-dont-even-need-antivirus-software-in-most-cases-saving-money-and-resources\" class=\"headerLink\"\u003e\n \u003ca href=\"#linuxs-package-management-repository-based-software-system-and-permission-model-all-contribute-to-its-strong-security-you-dont-even-need-antivirus-software-in-most-cases-saving-money-and-resources\" class=\"header-mark\" aria-label=\"Direct link to section: Every OS has its vulnerabilities, but Linux is known as one of the most secure operating systems. Windows systems are notorious for being targeted by malware and viruses, while Linux is inherently more secure — not invincible, but much harder to exploit. Linux’s package management, repository-based software system, and permission model all contribute to its strong security. You don’t even need antivirus software in most cases, saving money and resources.\"\u003e\u003c/a\u003eEvery OS has its vulnerabilities, but Linux is known as one of the \u003cstrong\u003emost secure operating systems\u003c/strong\u003e.\nWindows systems are notorious for being targeted by malware and viruses, while Linux is inherently more secure — not invincible, but \u003cstrong\u003emuch harder to exploit\u003c/strong\u003e.\nLinux’s package management, repository-based software system, and permission model all contribute to its strong security. You don’t even need antivirus software in most cases, saving money and resources.\u003c/h2\u003e\u003ch2 id=\"open-source\" class=\"headerLink\"\u003e\n \u003ca href=\"#open-source\" class=\"header-mark\" aria-label=\"Direct link to section: Open Source\"\u003e\u003c/a\u003eOpen Source\u003c/h2\u003e\u003ch2 id=\"even-if-most-users-wont-read-the-source-code-the-openness-ensures-transparency-and-trust-developers-can-inspect-modify-and-improve-linux-freely\" class=\"headerLink\"\u003e\n \u003ca href=\"#even-if-most-users-wont-read-the-source-code-the-openness-ensures-transparency-and-trust-developers-can-inspect-modify-and-improve-linux-freely\" class=\"header-mark\" aria-label=\"Direct link to section: Using a closed-source OS is like buying a car but never being allowed to open the hood. With Linux, you can see everything under the hood — it’s fully open source. Even if most users won’t read the source code, the openness ensures transparency and trust. Developers can inspect, modify, and improve Linux freely.\"\u003e\u003c/a\u003eUsing a closed-source OS is like buying a car but never being allowed to open the hood. With Linux, you can see everything under the hood — it’s \u003cstrong\u003efully open source\u003c/strong\u003e.\nEven if most users won’t read the source code, the openness ensures \u003cstrong\u003etransparency and trust\u003c/strong\u003e. Developers can inspect, modify, and improve Linux freely.\u003c/h2\u003e\u003ch2 id=\"lightweight-on-low-end-hardware\" class=\"headerLink\"\u003e\n \u003ca href=\"#lightweight-on-low-end-hardware\" class=\"header-mark\" aria-label=\"Direct link to section: Lightweight on Low-End Hardware\"\u003e\u003c/a\u003eLightweight on Low-End Hardware\u003c/h2\u003e\u003ch2 id=\"in-contrast-linux-can-run-smoothly-on-older-or-low-spec-computers-many-lightweight-distros-like-xubuntu-lubuntu-or-mx-linux-can-revive-old-machines-efficiently\" class=\"headerLink\"\u003e\n \u003ca href=\"#in-contrast-linux-can-run-smoothly-on-older-or-low-spec-computers-many-lightweight-distros-like-xubuntu-lubuntu-or-mx-linux-can-revive-old-machines-efficiently\" class=\"header-mark\" aria-label=\"Direct link to section: As operating systems evolve, hardware requirements also increase dramatically. For example, Windows 10 or 11 needs at least 8 GB of RAM for smooth performance, ideally 16 GB. In contrast, Linux can run smoothly on older or low-spec computers. Many lightweight distros like Xubuntu, Lubuntu, or MX Linux can revive old machines efficiently.\"\u003e\u003c/a\u003eAs operating systems evolve, hardware requirements also increase dramatically. For example, Windows 10 or 11 needs at least \u003cstrong\u003e8 GB of RAM\u003c/strong\u003e for smooth performance, ideally 16 GB.\nIn contrast, Linux can run smoothly on \u003cstrong\u003eolder or low-spec computers\u003c/strong\u003e. Many lightweight distros like \u003cstrong\u003eXubuntu, Lubuntu, or MX Linux\u003c/strong\u003e can revive old machines efficiently.\u003c/h2\u003e\u003ch2 id=\"highly-customizable\" class=\"headerLink\"\u003e\n \u003ca href=\"#highly-customizable\" class=\"header-mark\" aria-label=\"Direct link to section: Highly Customizable\"\u003e\u003c/a\u003eHighly Customizable\u003c/h2\u003e\u003cp\u003eCustomization is one of Linux’s biggest strengths. If you love to tweak your desktop’s look and feel, Linux is the best choice.\nDifferent desktop environments offer unique experiences:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eGNOME\u003c/strong\u003e – simple, elegant, and modern.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eKDE Plasma\u003c/strong\u003e – extremely customizable and full of features.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCinnamon, Budgie, Xfce\u003c/strong\u003e – great alternatives for different preferences.\nYou can even apply new icon themes, widgets, and wallpapers — or use \u003cstrong\u003eConky\u003c/strong\u003e to display real-time system info right on your desktop.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"perfect-for-programmers\" class=\"headerLink\"\u003e\n \u003ca href=\"#perfect-for-programmers\" class=\"header-mark\" aria-label=\"Direct link to section: Perfect for Programmers\"\u003e\u003c/a\u003ePerfect for Programmers\u003c/h2\u003e\u003ch2 id=\"moreover-linux-has-built-in-ssh-support-ideal-for-managing-servers-thats-why-most-developers-and-devops-professionals-prefer-linux-based-environments\" class=\"headerLink\"\u003e\n \u003ca href=\"#moreover-linux-has-built-in-ssh-support-ideal-for-managing-servers-thats-why-most-developers-and-devops-professionals-prefer-linux-based-environments\" class=\"header-mark\" aria-label=\"Direct link to section: Linux supports almost every major programming language (Python, C/C\u0026#43;\u0026#43;, Java, Perl, Ruby, etc.) and provides built-in development tools. The Linux terminal is far more powerful than Windows’ command prompt. Package managers like apt and dnf make software installation effortless. Moreover, Linux has built-in SSH support, ideal for managing servers. That’s why most developers and DevOps professionals prefer Linux-based environments.\"\u003e\u003c/a\u003eLinux supports almost every major programming language (Python, C/C++, Java, Perl, Ruby, etc.) and provides built-in development tools.\nThe \u003cstrong\u003eLinux terminal\u003c/strong\u003e is far more powerful than Windows’ command prompt. Package managers like \u003ccode\u003eapt\u003c/code\u003e and \u003ccode\u003ednf\u003c/code\u003e make software installation effortless.\nMoreover, Linux has built-in \u003cstrong\u003eSSH support\u003c/strong\u003e, ideal for managing servers. That’s why most developers and DevOps professionals prefer Linux-based environments.\u003c/h2\u003e\u003ch2 id=\"built-in-app-store\" class=\"headerLink\"\u003e\n \u003ca href=\"#built-in-app-store\" class=\"header-mark\" aria-label=\"Direct link to section: Built-in “App Store”\"\u003e\u003c/a\u003eBuilt-in “App Store”\u003c/h2\u003e\u003ch2 id=\"most-apps-youll-ever-need-are-available-directly-from-official-repositories-modern-distros-also-support-flatpak-snap-and-appimage-making-software-installation-consistent-and-reliable\" class=\"headerLink\"\u003e\n \u003ca href=\"#most-apps-youll-ever-need-are-available-directly-from-official-repositories-modern-distros-also-support-flatpak-snap-and-appimage-making-software-installation-consistent-and-reliable\" class=\"header-mark\" aria-label=\"Direct link to section: Installing software on Linux is easy thanks to Software Centers (like GNOME Software or KDE Discover). Most apps you’ll ever need are available directly from official repositories. Modern distros also support Flatpak, Snap, and AppImage, making software installation consistent and reliable.\"\u003e\u003c/a\u003eInstalling software on Linux is easy thanks to \u003cstrong\u003eSoftware Centers\u003c/strong\u003e (like GNOME Software or KDE Discover).\nMost apps you’ll ever need are available directly from official repositories. Modern distros also support \u003cstrong\u003eFlatpak, Snap, and AppImage\u003c/strong\u003e, making software installation consistent and reliable.\u003c/h2\u003e\u003ch2 id=\"many-distributions-to-choose-from\" class=\"headerLink\"\u003e\n \u003ca href=\"#many-distributions-to-choose-from\" class=\"header-mark\" aria-label=\"Direct link to section: Many Distributions to Choose From\"\u003e\u003c/a\u003eMany Distributions to Choose From\u003c/h2\u003e\u003cp\u003eUnlike Windows, Linux isn’t just one OS — it’s a whole family of \u003cstrong\u003edistributions (distros)\u003c/strong\u003e tailored to different needs.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eSecurity testing? Try \u003cstrong\u003eKali Linux\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eProgramming? Try \u003cstrong\u003eUbuntu\u003c/strong\u003e or \u003cstrong\u003eFedora\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eOld hardware? Try \u003cstrong\u003ePuppy Linux\u003c/strong\u003e or \u003cstrong\u003eLubuntu\u003c/strong\u003e.\nThere’s a Linux distro for everyone.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"reliability\" class=\"headerLink\"\u003e\n \u003ca href=\"#reliability\" class=\"header-mark\" aria-label=\"Direct link to section: Reliability\"\u003e\u003c/a\u003eReliability\u003c/h2\u003e\u003ch2 id=\"thats-why-linux-powers-most-of-the-worlds-web-servers-supercomputers-and-tech-giants-like-google-and-facebook\" class=\"headerLink\"\u003e\n \u003ca href=\"#thats-why-linux-powers-most-of-the-worlds-web-servers-supercomputers-and-tech-giants-like-google-and-facebook\" class=\"header-mark\" aria-label=\"Direct link to section: Windows tends to slow down over time, sometimes requiring a full reinstall. Linux, however, remains fast and stable for years. You rarely need to reboot Linux, except for kernel updates. With Windows, almost every small change requires a restart — install software, uninstall, update, reboot! That’s why Linux powers most of the world’s web servers, supercomputers, and tech giants like Google and Facebook.\"\u003e\u003c/a\u003eWindows tends to slow down over time, sometimes requiring a full reinstall. Linux, however, remains fast and stable for years.\nYou rarely need to reboot Linux, except for kernel updates. With Windows, almost every small change requires a restart — install software, uninstall, update, reboot!\nThat’s why Linux powers most of the world’s \u003cstrong\u003eweb servers, supercomputers, and tech giants\u003c/strong\u003e like Google and Facebook.\u003c/h2\u003e\u003ch2 id=\"supportive-community\" class=\"headerLink\"\u003e\n \u003ca href=\"#supportive-community\" class=\"header-mark\" aria-label=\"Direct link to section: Supportive Community\"\u003e\u003c/a\u003eSupportive Community\u003c/h2\u003e\u003ch2 id=\"just-search-forums-ask-on-telegram-groups-reddit-or-stackexchange--and-youll-likely-get-friendly-detailed-help-for-free\" class=\"headerLink\"\u003e\n \u003ca href=\"#just-search-forums-ask-on-telegram-groups-reddit-or-stackexchange--and-youll-likely-get-friendly-detailed-help-for-free\" class=\"header-mark\" aria-label=\"Direct link to section: You don’t need to hire a technician to fix Linux problems. The Linux community is incredibly active and helpful.\nJust search forums, ask on Telegram groups, Reddit, or StackExchange — and you’ll likely get friendly, detailed help for free.\"\u003e\u003c/a\u003eYou don’t need to hire a technician to fix Linux problems. The Linux community is incredibly active and helpful.\nJust search forums, ask on Telegram groups, Reddit, or StackExchange — and you’ll likely get friendly, detailed help for free.\u003c/h2\u003e\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003ch2 id=\"if-youre-curious-give-linux-a-try-you-might-just-find-your-own-reasons-to-stay\" class=\"headerLink\"\u003e\n \u003ca href=\"#if-youre-curious-give-linux-a-try-you-might-just-find-your-own-reasons-to-stay\" class=\"header-mark\" aria-label=\"Direct link to section: Those were my reasons for switching from Windows to Linux as my main operating system. Sure, Linux still has limitations — mainly with certain proprietary apps or games — but support is growing fast. More developers are creating native Linux versions every year. If you’re curious, give Linux a try. You might just find your own reasons to stay.\"\u003e\u003c/a\u003eThose were my \u003cstrong\u003ereasons for switching from Windows to Linux\u003c/strong\u003e as my main operating system.\nSure, Linux still has limitations — mainly with certain proprietary apps or games — but support is growing fast. More developers are creating native Linux versions every year.\nIf you’re curious, give Linux a try. You might just find your own reasons to stay.\u003c/h2\u003e\u003cp\u003e👉 So, are you ready to try Linux?\u003c/p\u003e", "language": "en" }, { "title" : "Desktop Environment", "date_published" : "2025-10-02T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/desktop-environment/", "url" : "https://www.aantriono.com/en/desktop-environment/", "summary": "This article discusses the desktop environments used across different Linux distributions.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the \u003cstrong\u003edesktop environments\u003c/strong\u003e used across different Linux distributions.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"overview\" class=\"headerLink\"\u003e\n \u003ca href=\"#overview\" class=\"header-mark\" aria-label=\"Direct link to section: Overview\"\u003e\u003c/a\u003eOverview\u003c/h2\u003e\u003cp\u003eFor Linux beginners, new terms can often feel overwhelming. One of the most common is the \u003cstrong\u003eDesktop Environment (DE)\u003c/strong\u003e.\nA \u003cstrong\u003eDesktop Environment\u003c/strong\u003e is a vital part of the Linux user experience. This article explains:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eWhat a desktop environment is and why it matters\u003c/li\u003e\n\u003cli\u003eThe main components of a DE\u003c/li\u003e\n\u003cli\u003ePopular desktop environments\u003c/li\u003e\n\u003cli\u003eVariants such as forks and spins\nUnderstanding these will help new users choose the right DE for their needs and preferences.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"what-is-a-desktop-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#what-is-a-desktop-environment\" class=\"header-mark\" aria-label=\"Direct link to section: What Is a Desktop Environment?\"\u003e\u003c/a\u003eWhat Is a Desktop Environment?\u003c/h2\u003e\u003cp\u003eA \u003cstrong\u003eDesktop Environment (DE)\u003c/strong\u003e is a collection of software that provides a \u003cstrong\u003eGraphical User Interface (GUI)\u003c/strong\u003e for interacting with the operating system.\nA DE typically includes:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eFile manager\u003c/li\u003e\n\u003cli\u003eIcons and themes\u003c/li\u003e\n\u003cli\u003eWallpaper and widgets\u003c/li\u003e\n\u003cli\u003ePanels and menus\u003c/li\u003e\n\u003cli\u003eBuilt-in applications (e.g., text editor, calculator, media player, web browser)\nMost Linux distributions come in two main editions:\u003c/li\u003e\n\u003c/ul\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eServer\u003c/strong\u003e (without GUI, command-line only)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDesktop\u003c/strong\u003e (with GUI provided by a DE)\nFor example, \u003cstrong\u003eUbuntu Desktop\u003c/strong\u003e ships with \u003cstrong\u003eGNOME\u003c/strong\u003e as its default DE. You can check which DE is currently running using:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003eecho\u003c/span\u003e \u003cspan class=\"nv\"\u003e$XDG_CURRENT_DESKTOP\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eExample output:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eubuntu:GNOME\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"without-a-de-interaction-is-limited-to-the-command-line-interface-cli-while-many-advanced-users-prefer-the-cli-a-de-makes-linux-more-approachable-for-newcomers\" class=\"headerLink\"\u003e\n \u003ca href=\"#without-a-de-interaction-is-limited-to-the-command-line-interface-cli-while-many-advanced-users-prefer-the-cli-a-de-makes-linux-more-approachable-for-newcomers\" class=\"header-mark\" aria-label=\"Direct link to section: Without a DE, interaction is limited to the Command Line Interface (CLI). While many advanced users prefer the CLI, a DE makes Linux more approachable for newcomers.\"\u003e\u003c/a\u003eWithout a DE, interaction is limited to the \u003cstrong\u003eCommand Line Interface (CLI)\u003c/strong\u003e. While many advanced users prefer the CLI, a DE makes Linux more approachable for newcomers.\u003c/h2\u003e\u003ch2 id=\"components-of-a-desktop-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#components-of-a-desktop-environment\" class=\"header-mark\" aria-label=\"Direct link to section: Components of a Desktop Environment\"\u003e\u003c/a\u003eComponents of a Desktop Environment\u003c/h2\u003e\u003cp\u003eA DE includes both visual and functional elements such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eFile manager\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePanels and menus\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThemes, icons, and fonts\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eWindow manager\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eApplication launcher\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTerminal emulator\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eVirtual workspaces\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSystem tray\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMouse pointer/cursor\u003c/strong\u003e\nIt also comes with built-in applications like:\u003c/li\u003e\n\u003cli\u003eText editor\u003c/li\u003e\n\u003cli\u003eCalculator\u003c/li\u003e\n\u003cli\u003eMedia player\u003c/li\u003e\n\u003cli\u003eWeb browser\u003c/li\u003e\n\u003cli\u003eDocument/PDF viewer\u003c/li\u003e\n\u003cli\u003eArchive manager\u003c/li\u003e\n\u003cli\u003eSystem monitor\nSimply put, everything that defines the look and feel of your Linux desktop is part of the \u003cstrong\u003eDesktop Environment\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"choosing-a-desktop-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#choosing-a-desktop-environment\" class=\"header-mark\" aria-label=\"Direct link to section: Choosing a Desktop Environment\"\u003e\u003c/a\u003eChoosing a Desktop Environment\u003c/h2\u003e\u003cp\u003eWith so many options available, choosing the right DE depends on your needs. Consider factors like:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eResource usage\u003c/strong\u003e (lightweight vs heavy)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAppearance\u003c/strong\u003e (classic, modern, or minimalist)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCustomization\u003c/strong\u003e (flexible vs simple)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePerformance\u003c/strong\u003e (optimized for old or new hardware)\nThere is no single “best” DE — only what fits you best. For example:\u003c/li\u003e\n\u003cli\u003eUsers with older hardware should consider \u003cstrong\u003eXfce\u003c/strong\u003e or \u003cstrong\u003eMATE\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eThose who prefer modern aesthetics may like \u003cstrong\u003eGNOME\u003c/strong\u003e or \u003cstrong\u003eBudgie\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003ePower users who enjoy heavy customization usually choose \u003cstrong\u003eKDE Plasma\u003c/strong\u003e.\nUltimately, pick the DE that feels comfortable and performs well on your system.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"popular-desktop-environments\" class=\"headerLink\"\u003e\n \u003ca href=\"#popular-desktop-environments\" class=\"header-mark\" aria-label=\"Direct link to section: Popular Desktop Environments\"\u003e\u003c/a\u003ePopular Desktop Environments\u003c/h2\u003e\u003ch3 id=\"gnome\" class=\"headerLink\"\u003e\n \u003ca href=\"#gnome\" class=\"header-mark\" aria-label=\"Direct link to section: GNOME\"\u003e\u003c/a\u003eGNOME\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePros:\u003c/strong\u003e Simple, intuitive, modern, and relatively resource-efficient.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBest for:\u003c/strong\u003e Beginners or those who want a clean, distraction-free interface.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDefault on:\u003c/strong\u003e Ubuntu, Fedora.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"kde-plasma\" class=\"headerLink\"\u003e\n \u003ca href=\"#kde-plasma\" class=\"header-mark\" aria-label=\"Direct link to section: KDE Plasma\"\u003e\u003c/a\u003eKDE Plasma\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePros:\u003c/strong\u003e Highly customizable, Windows-like interface, and more resource-efficient than older versions.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBest for:\u003c/strong\u003e Users who want total control over their desktop’s appearance and behavior.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDefault on:\u003c/strong\u003e Kubuntu, openSUSE.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"budgie\" class=\"headerLink\"\u003e\n \u003ca href=\"#budgie\" class=\"header-mark\" aria-label=\"Direct link to section: Budgie\"\u003e\u003c/a\u003eBudgie\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePros:\u003c/strong\u003e Minimalist, elegant, easy to use, with moderate customization options.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBest for:\u003c/strong\u003e Users who want a beautiful, modern desktop without complexity.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDefault on:\u003c/strong\u003e Solus, Ubuntu Budgie.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"xfce\" class=\"headerLink\"\u003e\n \u003ca href=\"#xfce\" class=\"header-mark\" aria-label=\"Direct link to section: Xfce\"\u003e\u003c/a\u003eXfce\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePros:\u003c/strong\u003e Extremely lightweight, stable, and efficient — perfect for older devices.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBest for:\u003c/strong\u003e Users seeking a fast, no-frills desktop experience.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDefault on:\u003c/strong\u003e Xubuntu, Manjaro Xfce.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"mate\" class=\"headerLink\"\u003e\n \u003ca href=\"#mate\" class=\"header-mark\" aria-label=\"Direct link to section: MATE\"\u003e\u003c/a\u003eMATE\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ePros:\u003c/strong\u003e Lightweight, traditional layout, stable — reminiscent of GNOME 2.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBest for:\u003c/strong\u003e Users who prefer a classic interface with great efficiency.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDefault on:\u003c/strong\u003e Ubuntu MATE, Linux Mint MATE.\nOther DEs include \u003cstrong\u003eCinnamon (Linux Mint)\u003c/strong\u003e, \u003cstrong\u003eLXQt/LXDE (lightweight)\u003c/strong\u003e, and \u003cstrong\u003ePantheon (elementary OS)\u003c/strong\u003e — each with unique strengths and visuals.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"desktop-environment-variants\" class=\"headerLink\"\u003e\n \u003ca href=\"#desktop-environment-variants\" class=\"header-mark\" aria-label=\"Direct link to section: Desktop Environment Variants\"\u003e\u003c/a\u003eDesktop Environment Variants\u003c/h2\u003e\u003cp\u003eBeyond official releases, many DEs have \u003cstrong\u003evariants\u003c/strong\u003e, known as \u003cstrong\u003eforks\u003c/strong\u003e or \u003cstrong\u003espins\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"fork\" class=\"headerLink\"\u003e\n \u003ca href=\"#fork\" class=\"header-mark\" aria-label=\"Direct link to section: Fork\"\u003e\u003c/a\u003eFork\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eA \u003cstrong\u003efork\u003c/strong\u003e is a derivative of an existing DE, often created to preserve old features or introduce new ones.\u003c/li\u003e\n\u003cli\u003eExample: \u003cstrong\u003eMATE\u003c/strong\u003e is a fork of \u003cstrong\u003eGNOME 2\u003c/strong\u003e, developed after GNOME 3’s release.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"spin\" class=\"headerLink\"\u003e\n \u003ca href=\"#spin\" class=\"header-mark\" aria-label=\"Direct link to section: Spin\"\u003e\u003c/a\u003eSpin\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eA \u003cstrong\u003espin\u003c/strong\u003e is an official edition of a Linux distribution featuring a different DE.\u003c/li\u003e\n\u003cli\u003eExample: \u003cstrong\u003eKubuntu\u003c/strong\u003e is an Ubuntu spin that uses \u003cstrong\u003eKDE Plasma\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eThe goal is to offer users a different DE experience without needing manual setup.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eA \u003cstrong\u003eDesktop Environment (DE)\u003c/strong\u003e is the graphical interface that allows users to interact visually with Linux.\u003c/li\u003e\n\u003cli\u003eWithout it, Linux is purely CLI-based.\u003c/li\u003e\n\u003cli\u003ePopular DEs include GNOME, KDE Plasma, Budgie, Xfce, MATE, Cinnamon, LXQt, and others.\u003c/li\u003e\n\u003cli\u003eEach DE has its advantages — choose one based on your hardware and preferences.\u003c/li\u003e\n\u003cli\u003eThe Linux ecosystem is further enriched by \u003cstrong\u003eforks\u003c/strong\u003e and \u003cstrong\u003espins\u003c/strong\u003e that offer flexibility and variety.\nBy understanding desktop environments, new users can better choose the right Linux experience — making learning Linux both enjoyable and productive.\u003c/li\u003e\n\u003c/ul\u003e", "language": "en" }, { "title" : "Writing Mathematical Expressions in Markdown", "date_published" : "2025-09-27T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/write-mathematical-expressions/", "url" : "https://www.aantriono.com/en/write-mathematical-expressions/", "summary": "This article discusses writing mathematical expressions in Markdown using KaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses writing mathematical expressions in Markdown using KaTeX.\u003c/p\u003e\n\u003cp\u003e\u0026ndash;\u003c/p\u003e\n\u003ch2 id=\"writing-mathematical-expressions-using-katex-in-markdown\" class=\"headerLink\"\u003e\n \u003ca href=\"#writing-mathematical-expressions-using-katex-in-markdown\" class=\"header-mark\" aria-label=\"Direct link to section: Writing Mathematical Expressions Using KaTeX in Markdown\"\u003e\u003c/a\u003eWriting Mathematical Expressions Using KaTeX in Markdown\u003c/h2\u003e\u003cp\u003eKaTeX is a JavaScript library that allows you to render mathematical expressions using LaTeX syntax on web pages. KaTeX is known for being fast and lightweight, making it ideal for displaying mathematical expressions in Markdown, especially in web or static site applications like Hugo, Jekyll, or plain HTML sites.\nThis article will explain how to write mathematical expressions using KaTeX in Markdown, including basic setup, supported LaTeX syntax, and several usage examples. With this guide, you can create Markdown documents that present mathematical expressions professionally.\u003c/p\u003e\n\u003ch2 id=\"why-katex\" class=\"headerLink\"\u003e\n \u003ca href=\"#why-katex\" class=\"header-mark\" aria-label=\"Direct link to section: Why KaTeX?\"\u003e\u003c/a\u003eWhy KaTeX?\u003c/h2\u003e\u003cp\u003eKaTeX has several advantages over other libraries like MathJax:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eSpeed: KaTeX is designed to render mathematical expressions instantly without affecting page performance.\u003c/li\u003e\n\u003cli\u003eSmall Size: KaTeX file size is smaller compared to other libraries.\u003c/li\u003e\n\u003cli\u003eMarkdown Compatibility: KaTeX is very suitable for use with Markdown on various platforms.\u003c/li\u003e\n\u003cli\u003eHigh-Quality Output: Mathematical expressions are rendered with a neat and aesthetic appearance.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"preparation-for-using-katex\" class=\"headerLink\"\u003e\n \u003ca href=\"#preparation-for-using-katex\" class=\"header-mark\" aria-label=\"Direct link to section: Preparation for Using KaTeX\"\u003e\u003c/a\u003ePreparation for Using KaTeX\u003c/h2\u003e\u003cp\u003eBefore you can write mathematical expressions with KaTeX in Markdown, you need to integrate KaTeX into your project.\u003c/p\u003e\n\u003ch2 id=\"installing-katex\" class=\"headerLink\"\u003e\n \u003ca href=\"#installing-katex\" class=\"header-mark\" aria-label=\"Direct link to section: Installing KaTeX\"\u003e\u003c/a\u003eInstalling KaTeX\u003c/h2\u003e\u003cp\u003eIf you\u0026rsquo;re working with static sites like Hugo or Jekyll, download KaTeX from its official website or use a CDN. Add KaTeX scripts and styles to your HTML file.\nExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 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\"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"cp\"\u003e\u0026lt;!DOCTYPE html\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ehtml\u003c/span\u003e \u003cspan class=\"na\"\u003elang\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;en\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ehead\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003emeta\u003c/span\u003e \u003cspan class=\"na\"\u003echarset\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;UTF-8\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003emeta\u003c/span\u003e \u003cspan class=\"na\"\u003ename\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;viewport\u0026#34;\u003c/span\u003e \u003cspan class=\"na\"\u003econtent\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;width=device-width, initial-scale=1.0\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etitle\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eKaTeX Example\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etitle\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003elink\u003c/span\u003e \u003cspan class=\"na\"\u003erel\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;stylesheet\u0026#34;\u003c/span\u003e \u003cspan class=\"na\"\u003ehref\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;https://cdn.jsdelivr.net/npm/katex@0.16.0/dist/katex.min.css\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e \u003cspan class=\"na\"\u003edefer\u003c/span\u003e \u003cspan class=\"na\"\u003esrc\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;https://cdn.jsdelivr.net/npm/katex@0.16.0/dist/katex.min.js\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e \u003cspan class=\"na\"\u003edefer\u003c/span\u003e \u003cspan class=\"na\"\u003esrc\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;https://cdn.jsdelivr.net/npm/katex@0.16.0/dist/contrib/auto-render.min.js\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003edocument\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"nx\"\u003eaddEventListener\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;DOMContentLoaded\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"kd\"\u003efunction\u003c/span\u003e \u003cspan class=\"p\"\u003e()\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003erenderMathInElement\u003c/span\u003e\u003cspan class=\"p\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003edocument\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"nx\"\u003ebody\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003edelimiters\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e \u003cspan class=\"nx\"\u003eleft\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;$$\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003eright\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;$$\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003edisplay\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"kc\"\u003etrue\u003c/span\u003e \u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e \u003cspan class=\"nx\"\u003eleft\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;$\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003eright\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;$\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003edisplay\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"kc\"\u003efalse\u003c/span\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e});\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e});\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ehead\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ebody\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e# KaTeX Example\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eThis is an example: $E = mc^2$\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ebody\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ehtml\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"integration-with-markdown-platforms\" class=\"headerLink\"\u003e\n \u003ca href=\"#integration-with-markdown-platforms\" class=\"header-mark\" aria-label=\"Direct link to section: Integration with Markdown Platforms\"\u003e\u003c/a\u003eIntegration with Markdown Platforms\u003c/h2\u003e\u003cp\u003eIf you\u0026rsquo;re using static site generators like Hugo or Jekyll, make sure to configure Markdown to support KaTeX. You can add KaTeX scripts and styles in the base layout (for example, baseof.html in Hugo).\u003c/p\u003e\n\u003ch2 id=\"latex-syntax-supported-by-katex\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-syntax-supported-by-katex\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Syntax Supported by KaTeX\"\u003e\u003c/a\u003eLaTeX Syntax Supported by KaTeX\u003c/h2\u003e\u003cp\u003eKaTeX supports most standard LaTeX/TeX syntax.\n\u003cdiv class=\"details admonition tip open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 352 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M96.06 454.35c.01 6.29 1.87 12.45 5.36 17.69l17.09 25.69a31.99 31.99 0 0 0 26.64 14.28h61.71a31.99 31.99 0 0 0 26.64-14.28l17.09-25.69a31.989 31.989 0 0 0 5.36-17.69l.04-38.35H96.01l.05 38.35zM0 176c0 44.37 16.45 84.85 43.56 115.78 16.52 18.85 42.36 58.23 52.21 91.45.04.26.07.52.11.78h160.24c.04-.26.07-.51.11-.78 9.85-33.22 35.69-72.6 52.21-91.45C335.55 260.85 352 220.37 352 176 352 78.61 272.91-.3 175.45 0 73.44.31 0 82.97 0 176zm176-80c-44.11 0-80 35.89-80 80 0 8.84-7.16 16-16 16s-16-7.16-16-16c0-61.76 50.24-112 112-112 8.84 0 16 7.16 16 16s-7.16 16-16 16z\"/\u003e\u003c/svg\u003e\u003c/span\u003eTip\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eHere is a list of \u003ca href=\"https://katex.org/docs/supported.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eTeX functions supported by KaTeX\u003c/a\u003e.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\nHere are some commonly used main categories:\u003c/p\u003e\n\u003ch3 id=\"a-inline-mathematics\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-inline-mathematics\" class=\"header-mark\" aria-label=\"Direct link to section: a. Inline Mathematics\"\u003e\u003c/a\u003ea. Inline Mathematics\u003c/h3\u003e\u003cp\u003eTo write inline mathematical expressions, use single dollar signs ($).\nExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$E = mc^2$\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$c = \\pm\\sqrt{a^2 + b^2}\\( and \\\\(f(x)=\\int_{-\\infty}^{\\infty} \\hat{f}(\\xi) e^{2 \\pi i \\xi x} d \\xi\\\\)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\n)E=mc^2\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003c/span\u003e\u003c/span\u003ec = \\pm\\sqrt{a^2 + b^2}\n \n \\( and \\\\(f(x)=\\int_{-\\infty}^{\\infty} \\hat{f}(\\xi) e^{2 \\pi i \\xi x} d \\xi\\\\)\u003c/p\u003e\n\u003ch3 id=\"b-block-mathematics\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-block-mathematics\" class=\"header-mark\" aria-label=\"Direct link to section: b. Block Mathematics\"\u003e\u003c/a\u003eb. Block Mathematics\u003c/h3\u003e\u003cp\u003eUse double dollar signs ($$) to create mathematical expressions in block form.\nExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\\int_a^b f(x) \\ dx = F(b) - F(a)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmo\u003e∫\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msubsup\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\int_a^b f(x) \\ dx = F(b) - F(a)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.511em;vertical-align:-0.9119em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003e\u003cspan class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\"\u003e∫\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.599em;\"\u003e\u003cspan style=\"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8129em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9119em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"c-mathematical-operators\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-mathematical-operators\" class=\"header-mark\" aria-label=\"Direct link to section: c. Mathematical Operators\"\u003e\u003c/a\u003ec. Mathematical Operators\u003c/h3\u003e\u003cp\u003eKaTeX supports various mathematical operators, including:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eAddition: x + y\u003c/li\u003e\n\u003cli\u003eMultiplication: x \\cdot y or x \\times y\u003c/li\u003e\n\u003cli\u003eFractions: \\frac{a}{b}\u003c/li\u003e\n\u003cli\u003eRoots: \\sqrt{x}\nExample:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}}\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mfrac\u003e\u003cmsup\u003e\u003cmi\u003ee\u003c/mi\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2514em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ee\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.206em;\"\u003e\u003cspan style=\"top:-3.413em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1329em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0484em;\"\u003e\u003cspan style=\"top:-3.0484em;margin-right:0.1em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"d-greek-and-other-symbols\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-greek-and-other-symbols\" class=\"header-mark\" aria-label=\"Direct link to section: d. Greek and Other Symbols\"\u003e\u003c/a\u003ed. Greek and Other Symbols\u003c/h3\u003e\u003cp\u003eUse LaTeX syntax to write Greek or special symbols:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eLowercase Greek: \\alpha, \\beta, \\gamma\u003c/li\u003e\n\u003cli\u003eUppercase Greek: \\Gamma, \\Delta\u003c/li\u003e\n\u003cli\u003eOthers: \\infty (infinity), \\partial (partial derivative), \\nabla (nabla operator)\nExample:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\\)\\alpha + \\beta = \\gamma\\(\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eγ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha + \\beta = \\gamma\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05556em;\"\u003eγ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"e-matrices-and-systems-of-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#e-matrices-and-systems-of-equations\" class=\"header-mark\" aria-label=\"Direct link to section: e. Matrices and Systems of Equations\"\u003e\u003c/a\u003ee. Matrices and Systems of Equations\u003c/h3\u003e\u003cp\u003eKaTeX supports matrix format using the \\begin{matrix} command.\nExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\\begin{bmatrix}\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ea \u003cspan class=\"err\"\u003e\u0026amp;\u003c/span\u003e b \\\\\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ec \u003cspan class=\"err\"\u003e\u0026amp;\u003c/span\u003e d\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\\end{bmatrix}\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e[\u003c/mo\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cmo fence=\"true\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{bmatrix}\na \u0026amp; b \\\\\nc \u0026amp; d\n\\end{bmatrix}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e[\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.45em;\"\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.95em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.45em;\"\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.95em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"further-customization\" class=\"headerLink\"\u003e\n \u003ca href=\"#further-customization\" class=\"header-mark\" aria-label=\"Direct link to section: Further Customization\"\u003e\u003c/a\u003eFurther Customization\u003c/h2\u003e\u003ch3 id=\"a-custom-delimiters\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-custom-delimiters\" class=\"header-mark\" aria-label=\"Direct link to section: a. Custom Delimiters\"\u003e\u003c/a\u003ea. Custom Delimiters\u003c/h3\u003e\u003cp\u003eYou can customize delimiters in KaTeX. For example, use \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e and \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ean\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e for inline, or [] for blocks.\u003c/p\u003e\n\u003ch3 id=\"b-visual-styles\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-visual-styles\" class=\"header-mark\" aria-label=\"Direct link to section: b. Visual Styles\"\u003e\u003c/a\u003eb. Visual Styles\u003c/h3\u003e\u003cp\u003eKaTeX supports visual styles such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eBold Text: \\mathbf{A} for bold letters.\u003c/li\u003e\n\u003cli\u003eItalic Letters: \\mathit{A} for italic letters.\u003c/li\u003e\n\u003cli\u003eFont Size: \\small, \\large, \\Huge.\nExample:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\\mathbf{F} = \\frac{d}{dt} \\mathit{p}\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e$$\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"bold\"\u003eF\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\mathbf{F} = \\frac{d}{dt} \\mathit{p}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathbf\"\u003eF\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathit\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"c-error-handling\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-error-handling\" class=\"header-mark\" aria-label=\"Direct link to section: c. Error Handling\"\u003e\u003c/a\u003ec. Error Handling\u003c/h3\u003e\u003cp\u003eKaTeX has good error handling. If you write incorrect syntax, the expression will still be rendered but will provide an error notification.\u003c/p\u003e\n\u003ch2 id=\"mathematical-expressions-with-katex\" class=\"headerLink\"\u003e\n \u003ca href=\"#mathematical-expressions-with-katex\" class=\"header-mark\" aria-label=\"Direct link to section: Mathematical Expressions with KaTeX\"\u003e\u003c/a\u003eMathematical Expressions with KaTeX\u003c/h2\u003e\u003cp\u003eThis is an example of using KaTeX in Markdown.\u003c/p\u003e\n\u003ch2 id=\"inline-example\" class=\"headerLink\"\u003e\n \u003ca href=\"#inline-example\" class=\"header-mark\" aria-label=\"Direct link to section: Inline Example\"\u003e\u003c/a\u003eInline Example\u003c/h2\u003e\u003cp\u003eFamous formula by Einstein: )E = mc^2(\u003c/p\u003e\n\u003ch2 id=\"block-example\" class=\"headerLink\"\u003e\n \u003ca href=\"#block-example\" class=\"header-mark\" aria-label=\"Direct link to section: Block Example\"\u003e\u003c/a\u003eBlock Example\u003c/h2\u003e\u003cp\u003eHere is an integral:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmo\u003e∫\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msubsup\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\int_a^b f(x) \\, dx = F(b) - F(a)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.511em;vertical-align:-0.9119em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003e\u003cspan class=\"mop op-symbol large-op\" style=\"margin-right:0.44445em;position:relative;top:-0.0011em;\"\u003e∫\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.599em;\"\u003e\u003cspan style=\"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8129em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9119em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"matrix\" class=\"headerLink\"\u003e\n \u003ca href=\"#matrix\" class=\"header-mark\" aria-label=\"Direct link to section: Matrix\"\u003e\u003c/a\u003eMatrix\u003c/h2\u003e\u003cp\u003eMatrix representation:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e[\u003c/mo\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cmo fence=\"true\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{bmatrix}\n1 \u0026amp; 2 \\\\\n3 \u0026amp; 4\n\\end{bmatrix}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e[\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.45em;\"\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.95em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.45em;\"\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.95em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"greek-symbols\" class=\"headerLink\"\u003e\n \u003ca href=\"#greek-symbols\" class=\"header-mark\" aria-label=\"Direct link to section: Greek Symbols\"\u003e\u003c/a\u003eGreek Symbols\u003c/h2\u003e\u003cp\u003eSome Greek symbols: )\\alpha\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e, \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.3em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\\beta\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e, \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.3em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\\gamma$.\u003c/p\u003e\n\u003ch2 id=\"visual-styles\" class=\"headerLink\"\u003e\n \u003ca href=\"#visual-styles\" class=\"header-mark\" aria-label=\"Direct link to section: Visual Styles\"\u003e\u003c/a\u003eVisual Styles\u003c/h2\u003e\u003cp\u003eBold letters: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"bold\"\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\mathbf{A}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathbf\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\nItalic letters: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\mathit{A}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathit\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\nThe results will be rendered according to your KaTeX configuration.\u003c/p\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eWith KaTeX, you can create beautiful and efficient mathematical expressions in Markdown.\n\u003cdiv class=\"details admonition tip open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 352 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M96.06 454.35c.01 6.29 1.87 12.45 5.36 17.69l17.09 25.69a31.99 31.99 0 0 0 26.64 14.28h61.71a31.99 31.99 0 0 0 26.64-14.28l17.09-25.69a31.989 31.989 0 0 0 5.36-17.69l.04-38.35H96.01l.05 38.35zM0 176c0 44.37 16.45 84.85 43.56 115.78 16.52 18.85 42.36 58.23 52.21 91.45.04.26.07.52.11.78h160.24c.04-.26.07-.51.11-.78 9.85-33.22 35.69-72.6 52.21-91.45C335.55 260.85 352 220.37 352 176 352 78.61 272.91-.3 175.45 0 73.44.31 0 82.97 0 176zm176-80c-44.11 0-80 35.89-80 80 0 8.84-7.16 16-16 16s-16-7.16-16-16c0-61.76 50.24-112 112-112 8.84 0 16 7.16 16 16s-7.16 16-16 16z\"/\u003e\u003c/svg\u003e\u003c/span\u003eTip\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eIf you want to learn more about LaTeX or TeX, you can visit \u003ca href=\"https://www.aantriono.com/2022/07/buku-panduan-belajar-latex.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehere\u003c/a\u003e. Check it out directly. 😙\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\nSimple setup steps and extensive LaTeX syntax support make KaTeX an excellent choice for online mathematical needs. Make sure to adjust settings according to your project so that KaTeX runs optimally.\u003c/p\u003e", "language": "en" }, { "title" : "Basic Markdown Syntax", "date_published" : "2025-09-01T21:57:40+08:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/basic-markdown-syntax/", "url" : "https://www.aantriono.com/en/basic-markdown-syntax/", "summary": "This article discusses examples of basic Markdown syntax that can be used in Hugo content files.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses examples of basic Markdown syntax that can be used in Hugo content files.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eThis article is a copy from the \u003ca href=\"https://learn.getgrav.org/content/markdown\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eGrav original\u003c/a\u003e page.\nIf you want to know more comprehensively, please read the \u003ca href=\"../theme-documentation-content#extended-markdown-syntax\" rel=\"\"\u003eMarkdown Syntax\u003c/a\u003e page.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\nWriting content for the Web is tedious. WYSIWYG editors help alleviate this task, but they generally result in messy code, or worse yet, ugly web pages.\nMarkdown is a better way to write HTML, without all the complexities and ugliness that usually accompanies it.\nSome of the key benefits of Markdown are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eMarkdown is easy to learn, with minimal extra characters, so it\u0026rsquo;s also quicker to write content.\u003c/li\u003e\n\u003cli\u003eLess chance of errors when writing in Markdown.\u003c/li\u003e\n\u003cli\u003eProduces valid XHTML output.\u003c/li\u003e\n\u003cli\u003eKeeps content and visual display separate, so you cannot mess up the look of your site.\u003c/li\u003e\n\u003cli\u003eWrite in any text editor or Markdown application you like.\nJohn Gruber, the author of Markdown, puts it like this:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cblockquote\u003e\n \u003cp\u003eThe overriding design goal for Markdown\u0026rsquo;s formatting syntax is to make it as readable as possible.\nThe idea is that a Markdown-formatted document should be publishable as-is, as plain text, without looking like it\u0026rsquo;s been marked up with tags or formatting instructions.\nWhile Markdown\u0026rsquo;s syntax has been influenced by several existing text-to-HTML filters, the single biggest source of inspiration for Markdown\u0026rsquo;s syntax is the format of plain text email.\u003c/p\u003e\n\u003cp\u003e\u003cdiv id=\"id-43\"\u003e\u0026ndash; \u003cem\u003eJohn Gruber\u003c/em\u003e\u003c/div\u003e\nWithout further delay, let us go over the main elements of Markdown and what the resulting HTML looks like!\n\u003cdiv class=\"details admonition tip open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 352 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M96.06 454.35c.01 6.29 1.87 12.45 5.36 17.69l17.09 25.69a31.99 31.99 0 0 0 26.64 14.28h61.71a31.99 31.99 0 0 0 26.64-14.28l17.09-25.69a31.989 31.989 0 0 0 5.36-17.69l.04-38.35H96.01l.05 38.35zM0 176c0 44.37 16.45 84.85 43.56 115.78 16.52 18.85 42.36 58.23 52.21 91.45.04.26.07.52.11.78h160.24c.04-.26.07-.51.11-.78 9.85-33.22 35.69-72.6 52.21-91.45C335.55 260.85 352 220.37 352 176 352 78.61 272.91-.3 175.45 0 73.44.31 0 82.97 0 176zm176-80c-44.11 0-80 35.89-80 80 0 8.84-7.16 16-16 16s-16-7.16-16-16c0-61.76 50.24-112 112-112 8.84 0 16 7.16 16 16s-7.16 16-16 16z\"/\u003e\u003c/svg\u003e\u003c/span\u003eTip\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u0026#x1f604; Bookmark this page for easy future reference!\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"headings\" class=\"headerLink\"\u003e\n \u003ca href=\"#headings\" class=\"header-mark\" aria-label=\"Direct link to section: Headings\"\u003e\u003c/a\u003eHeadings\u003c/h2\u003e\u003cp\u003eHeadings from \u003ccode\u003eh2\u003c/code\u003e through \u003ccode\u003eh6\u003c/code\u003e are constructed with a \u003ccode\u003e#\u003c/code\u003e for each level:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e## h2 Heading\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e### h3 Heading\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e#### h4 Heading\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e##### h5 Heading\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e\u003c/span\u003e###### h6 Heading\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe HTML output looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eh2\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eh2 Heading\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eh2\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eh3\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eh3 Heading\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eh3\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eh4\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eh4 Heading\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eh4\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eh5\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eh5 Heading\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eh5\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eh6\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eh6 Heading\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eh6\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eHeading IDs\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cp\u003eTo add a custom heading ID, enclose the custom ID in curly braces on the same line as the heading:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-44\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e### A Great Heading {\u003cspan class=\"ni\"\u003e#custom\u003c/span\u003e-id}\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe HTML output looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-45\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eh3\u003c/span\u003e \u003cspan class=\"na\"\u003eid\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;custom-id\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eA Great Heading\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eh3\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch2 id=\"comments\" class=\"headerLink\"\u003e\n \u003ca href=\"#comments\" class=\"header-mark\" aria-label=\"Direct link to section: Comments\"\u003e\u003c/a\u003eComments\u003c/h2\u003e\u003cp\u003eComments should be HTML compatible.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u0026lt;!--\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003eThis is a comment\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e--\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eComment below should \u003cstrong\u003eNOT\u003c/strong\u003e be seen:\u003c/p\u003e\n\u003c!--\nThis is a comment\n--\u003e\n\u003ch2 id=\"horizontal-rules\" class=\"headerLink\"\u003e\n \u003ca href=\"#horizontal-rules\" class=\"header-mark\" aria-label=\"Direct link to section: Horizontal Rules\"\u003e\u003c/a\u003eHorizontal Rules\u003c/h2\u003e\u003cp\u003eThe HTML \u003ccode\u003e\u0026lt;hr\u0026gt;\u003c/code\u003e element is for creating a \u0026ldquo;thematic break\u0026rdquo; between paragraph-level elements. In Markdown, you can create a \u003ccode\u003e\u0026lt;hr\u0026gt;\u003c/code\u003e with any of the following:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e___\u003c/code\u003e: three consecutive underscores\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e---\u003c/code\u003e: three consecutive dashes\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e***\u003c/code\u003e: three consecutive asterisks\nThe rendered output looks like this:\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003chr\u003e\n\u003chr\u003e\n\u003ch2 id=\"body-copy\" class=\"headerLink\"\u003e\n \u003ca href=\"#body-copy\" class=\"header-mark\" aria-label=\"Direct link to section: Body Copy\"\u003e\u003c/a\u003eBody Copy\u003c/h2\u003e\u003cp\u003eBody copy is written as normal, plain text and will be wrapped with `` tags in the rendered HTML.\nSo this body copy:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eLorem ipsum dolor sit amet, graecis denique ei vel, at duo primis mandamus. Et legere ocurreret pri,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eanimal tacimates complectitur ad cum. Cu eum inermis inimicus efficiendi. Labore officiis his ex,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esoluta officiis concludaturque ei qui, vide sensibus vim ad.\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIn HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eLorem ipsum dolor sit amet, graecis denique ei vel, at duo primis mandamus. Et legere ocurreret pri, animal tacimates complectitur ad cum. Cu eum inermis inimicus efficiendi. Labore officiis his ex, soluta officiis concludaturque ei qui, vide sensibus vim ad.\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eA \u003cstrong\u003eline break\u003c/strong\u003e can be done with one blank line.\u003c/p\u003e\n\u003ch2 id=\"inline-html\" class=\"headerLink\"\u003e\n \u003ca href=\"#inline-html\" class=\"header-mark\" aria-label=\"Direct link to section: Inline HTML\"\u003e\u003c/a\u003eInline HTML\u003c/h2\u003e\u003cp\u003eIf you need a certain HTML tag (with a class), you can simply use HTML:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eParagraph in Markdown.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ediv\u003c/span\u003e \u003cspan class=\"na\"\u003eclass\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;class\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e This is \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eb\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eHTML\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eb\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ediv\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eParagraph in Markdown.\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"emphasis\" class=\"headerLink\"\u003e\n \u003ca href=\"#emphasis\" class=\"header-mark\" aria-label=\"Direct link to section: Emphasis\"\u003e\u003c/a\u003eEmphasis\u003c/h2\u003e\u003ch3 id=\"bold\" class=\"headerLink\"\u003e\n \u003ca href=\"#bold\" class=\"header-mark\" aria-label=\"Direct link to section: Bold\"\u003e\u003c/a\u003eBold\u003c/h3\u003e\u003cp\u003eFor emphasizing a snippet of text with a heavier font weight.\nThe following snippet of text is \u003cstrong\u003erendered as bold text\u003c/strong\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e**rendered as bold text**\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e__rendered as bold text__\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003erendered as bold text\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"italics\" class=\"headerLink\"\u003e\n \u003ca href=\"#italics\" class=\"header-mark\" aria-label=\"Direct link to section: Italics\"\u003e\u003c/a\u003eItalics\u003c/h3\u003e\u003cp\u003eFor emphasizing a snippet of text with italics.\nThe following snippet of text is \u003cem\u003erendered as italicized text\u003c/em\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e*rendered as italicized text*\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e_rendered as italicized text_\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003erendered as italicized text\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"strikethrough\" class=\"headerLink\"\u003e\n \u003ca href=\"#strikethrough\" class=\"header-mark\" aria-label=\"Direct link to section: Strikethrough\"\u003e\u003c/a\u003eStrikethrough\u003c/h3\u003e\u003cp\u003eIn \u003ca href=\"https://github.github.com/gfm/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003e[GFM]^(GitHub flavored Markdown)\u003c/a\u003e you can do strikethroughs.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e~~Strike through this text.~~\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\n\u003cdel\u003eStrike through this text.\u003c/del\u003e\nThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eStrike through this text.\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"combination\" class=\"headerLink\"\u003e\n \u003ca href=\"#combination\" class=\"header-mark\" aria-label=\"Direct link to section: Combination\"\u003e\u003c/a\u003eCombination\u003c/h3\u003e\u003cp\u003eBold, italics, and strikethrough can be used in combination:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e***bold and italics***\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gd\"\u003e~~**strikethrough and bold**~~\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gd\"\u003e~~*strikethrough and italics*~~\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e~~***bold, italics and strikethrough***~~\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\n\u003cem\u003e\u003cstrong\u003ebold and italics\u003c/strong\u003e\u003c/em\u003e\n\u003cdel\u003e\u003cstrong\u003estrikethrough and bold\u003c/strong\u003e\u003c/del\u003e\n\u003cdel\u003e\u003cem\u003estrikethrough and italics\u003c/em\u003e\u003c/del\u003e\n\u003cdel\u003e\u003cem\u003e\u003cstrong\u003ebold, italics and strikethrough\u003c/strong\u003e\u003c/em\u003e\u003c/del\u003e\nThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003ebold and italics\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003estrikethrough and bold\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003estrikethrough and italics\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003ebold, italics and strikethrough\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eem\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003edel\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"blockquotes\" class=\"headerLink\"\u003e\n \u003ca href=\"#blockquotes\" class=\"header-mark\" aria-label=\"Direct link to section: Blockquotes\"\u003e\u003c/a\u003eBlockquotes\u003c/h2\u003e\u003cp\u003eFor quoting blocks of content from another source within your document.\nAdd \u003ccode\u003e\u0026gt;\u003c/code\u003e before any text you want to quote:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u0026gt; \u003cspan class=\"gs\"\u003e**Fusion Drive**\u003c/span\u003e combines a hard drive with a flash storage (solid-state drive) and presents it as a single logical volume with the space of both drives combined.\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eFusion Drive\u003c/strong\u003e combines a hard drive with a flash storage (solid-state drive) and presents it as a single logical volume with the space of both drives combined.\nThe HTML looks like this:\u003c/p\u003e\n\u003c/blockquote\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-16\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eblockquote\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eFusion Drive\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e combines a hard drive with a flash storage (solid-state drive) and presents it as a single logical volume with the space of both drives combined.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eblockquote\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eBlockquotes can also be nested:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-17\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\u0026gt; \u003c/span\u003e\u003cspan class=\"ge\"\u003eDonec massa lacus, ultricies a ullamcorper in, fermentum sed augue.\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"ge\"\u003e\u003c/span\u003eNunc augue augue, aliquam non hendrerit ac, commodo vel nisi.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u0026gt;\u0026gt; Sed adipiscing elit vitae augue consectetur a gravida nunc vehicula. Donec auctor\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eodio non est accumsan facilisis. Aliquam id turpis in dolor tincidunt mollis ac eu diam.\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eDonec massa lacus, ultricies a ullamcorper in, fermentum sed augue.\nNunc augue augue, aliquam non hendrerit ac, commodo vel nisi.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eSed adipiscing elit vitae augue consectetur a gravida nunc vehicula. Donec auctor\nodio non est accumsan facilisis. Aliquam id turpis in dolor tincidunt mollis ac eu diam.\u003c/p\u003e\n\u003c/blockquote\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"lists\" class=\"headerLink\"\u003e\n \u003ca href=\"#lists\" class=\"header-mark\" aria-label=\"Direct link to section: Lists\"\u003e\u003c/a\u003eLists\u003c/h2\u003e\u003ch3 id=\"unordered\" class=\"headerLink\"\u003e\n \u003ca href=\"#unordered\" class=\"header-mark\" aria-label=\"Direct link to section: Unordered\"\u003e\u003c/a\u003eUnordered\u003c/h3\u003e\u003cp\u003eA list of items in which the order of the items does not explicitly matter.\nYou may use any of the following symbols to denote bullets for each list item:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-18\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e valid bullet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e-\u003c/span\u003e valid bullet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e+ valid bullet\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eFor example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-19\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e Lorem ipsum dolor sit amet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e Consectetur adipiscing elit\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e Integer molestie lorem at massa\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e Facilisis in pretium nisl aliquet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e Nulla volutpat aliquam velit\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e*\u003c/span\u003e Phasellus iaculis neque\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e*\u003c/span\u003e Purus sodales ultricies\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e*\u003c/span\u003e Vestibulum laoreet porttitor sem\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e*\u003c/span\u003e Ac tristique libero volutpat at\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e Faucibus porta lacus fringilla vel\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e*\u003c/span\u003e Aenean sit amet erat nunc\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e* Eget porttitor lorem\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eLorem ipsum dolor sit amet\u003c/li\u003e\n\u003cli\u003eConsectetur adipiscing elit\u003c/li\u003e\n\u003cli\u003eInteger molestie lorem at massa\u003c/li\u003e\n\u003cli\u003eFacilisis in pretium nisl aliquet\u003c/li\u003e\n\u003cli\u003eNulla volutpat aliquam velit\n\u003cul\u003e\n\u003cli\u003ePhasellus iaculis neque\u003c/li\u003e\n\u003cli\u003ePurus sodales ultricies\u003c/li\u003e\n\u003cli\u003eVestibulum laoreet porttitor sem\u003c/li\u003e\n\u003cli\u003eAc tristique libero volutpat at\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003eFaucibus porta lacus fringilla vel\u003c/li\u003e\n\u003cli\u003eAenean sit amet erat nunc\u003c/li\u003e\n\u003cli\u003eEget porttitor lorem\nThe HTML looks like this:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-20\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eul\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eLorem ipsum dolor sit amet\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eConsectetur adipiscing elit\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eInteger molestie lorem at massa\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eFacilisis in pretium nisl aliquet\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eNulla volutpat aliquam velit\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eul\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003ePhasellus iaculis neque\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003ePurus sodales ultricies\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eVestibulum laoreet porttitor sem\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eAc tristique libero volutpat at\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eul\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eFaucibus porta lacus fringilla vel\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eAenean sit amet erat nunc\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eEget porttitor lorem\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eul\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"ordered\" class=\"headerLink\"\u003e\n \u003ca href=\"#ordered\" class=\"header-mark\" aria-label=\"Direct link to section: Ordered\"\u003e\u003c/a\u003eOrdered\u003c/h3\u003e\u003cp\u003eA list of items in which the order of items does explicitly matter.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-21\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Lorem ipsum dolor sit amet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e2.\u003c/span\u003e Consectetur adipiscing elit\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e3.\u003c/span\u003e Integer molestie lorem at massa\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e4.\u003c/span\u003e Facilisis in pretium nisl aliquet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e5.\u003c/span\u003e Nulla volutpat aliquam velit\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e6.\u003c/span\u003e Faucibus porta lacus fringilla vel\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e7.\u003c/span\u003e Aenean sit amet erat nunc\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e8. Eget porttitor lorem\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eLorem ipsum dolor sit amet\u003c/li\u003e\n\u003cli\u003eConsectetur adipiscing elit\u003c/li\u003e\n\u003cli\u003eInteger molestie lorem at massa\u003c/li\u003e\n\u003cli\u003eFacilisis in pretium nisl aliquet\u003c/li\u003e\n\u003cli\u003eNulla volutpat aliquam velit\u003c/li\u003e\n\u003cli\u003eFaucibus porta lacus fringilla vel\u003c/li\u003e\n\u003cli\u003eAenean sit amet erat nunc\u003c/li\u003e\n\u003cli\u003eEget porttitor lorem\nThe HTML looks like this:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-22\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eol\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eLorem ipsum dolor sit amet\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eConsectetur adipiscing elit\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eInteger molestie lorem at massa\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eFacilisis in pretium nisl aliquet\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eNulla volutpat aliquam velit\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eFaucibus porta lacus fringilla vel\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eAenean sit amet erat nunc\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eEget porttitor lorem\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eli\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eol\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cdiv class=\"details admonition tip open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 352 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M96.06 454.35c.01 6.29 1.87 12.45 5.36 17.69l17.09 25.69a31.99 31.99 0 0 0 26.64 14.28h61.71a31.99 31.99 0 0 0 26.64-14.28l17.09-25.69a31.989 31.989 0 0 0 5.36-17.69l.04-38.35H96.01l.05 38.35zM0 176c0 44.37 16.45 84.85 43.56 115.78 16.52 18.85 42.36 58.23 52.21 91.45.04.26.07.52.11.78h160.24c.04-.26.07-.51.11-.78 9.85-33.22 35.69-72.6 52.21-91.45C335.55 260.85 352 220.37 352 176 352 78.61 272.91-.3 175.45 0 73.44.31 0 82.97 0 176zm176-80c-44.11 0-80 35.89-80 80 0 8.84-7.16 16-16 16s-16-7.16-16-16c0-61.76 50.24-112 112-112 8.84 0 16 7.16 16 16s-7.16 16-16 16z\"/\u003e\u003c/svg\u003e\u003c/span\u003eTip\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cp\u003eIf you just use \u003ccode\u003e1.\u003c/code\u003e for each number, Markdown will automatically number each item. For example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-46\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Lorem ipsum dolor sit amet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Consectetur adipiscing elit\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Integer molestie lorem at massa\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Facilisis in pretium nisl aliquet\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Nulla volutpat aliquam velit\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Faucibus porta lacus fringilla vel\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e1.\u003c/span\u003e Aenean sit amet erat nunc\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1. Eget porttitor lorem\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eLorem ipsum dolor sit amet\u003c/li\u003e\n\u003cli\u003eConsectetur adipiscing elit\u003c/li\u003e\n\u003cli\u003eInteger molestie lorem at massa\u003c/li\u003e\n\u003cli\u003eFacilisis in pretium nisl aliquet\u003c/li\u003e\n\u003cli\u003eNulla volutpat aliquam velit\u003c/li\u003e\n\u003cli\u003eFaucibus porta lacus fringilla vel\u003c/li\u003e\n\u003cli\u003eAenean sit amet erat nunc\u003c/li\u003e\n\u003cli\u003eEget porttitor lorem\u003c/li\u003e\n\u003c/ol\u003e\n\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch3 id=\"task-lists\" class=\"headerLink\"\u003e\n \u003ca href=\"#task-lists\" class=\"header-mark\" aria-label=\"Direct link to section: Task Lists\"\u003e\u003c/a\u003eTask Lists\u003c/h3\u003e\u003cp\u003eTask lists allow you to create a list of items with checkboxes. To create a task list, add dashes (\u003ccode\u003e-\u003c/code\u003e) and brackets with a space (\u003ccode\u003e[ ]\u003c/code\u003e) before task list items. To select a checkbox, add an x between the brackets (\u003ccode\u003e[x]\u003c/code\u003e).\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-23\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e- [x]\u003c/span\u003e Write the press release\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e- [ ]\u003c/span\u003e Update the website\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e- [ ] Contact the media\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cinput checked=\"\" disabled=\"\" type=\"checkbox\"\u003e Write the press release\u003c/li\u003e\n\u003cli\u003e\u003cinput disabled=\"\" type=\"checkbox\"\u003e Update the website\u003c/li\u003e\n\u003cli\u003e\u003cinput disabled=\"\" type=\"checkbox\"\u003e Contact the media\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"code\" class=\"headerLink\"\u003e\n \u003ca href=\"#code\" class=\"header-mark\" aria-label=\"Direct link to section: Code\"\u003e\u003c/a\u003eCode\u003c/h2\u003e\u003ch3 id=\"inline-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#inline-code\" class=\"header-mark\" aria-label=\"Direct link to section: Inline Code\"\u003e\u003c/a\u003eInline Code\u003c/h3\u003e\u003cp\u003eWrap inline snippets of code with \u003ccode\u003e\u0026lt;code\u0026gt;\u0026lt;/code\u0026gt;\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-24\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eIn this example, \u003cspan class=\"sb\"\u003e`\u0026lt;section\u0026gt;\u0026lt;/section\u0026gt;`\u003c/span\u003e should be wrapped as \u003cspan class=\"gs\"\u003e**code**\u003c/span\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\nIn this example, \u003ccode\u003e\u0026lt;section\u0026gt;\u0026lt;/section\u0026gt;\u003c/code\u003e should be wrapped as \u003cstrong\u003ecode\u003c/strong\u003e.\nThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-25\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e In this example, \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ecode\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"ni\"\u003e\u0026amp;lt;\u003c/span\u003esection\u003cspan class=\"ni\"\u003e\u0026amp;gt;\u0026amp;lt;\u003c/span\u003e/section\u003cspan class=\"ni\"\u003e\u0026amp;gt;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ecode\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e should be wrapped with \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003ecode\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003estrong\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ep\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"indented-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#indented-code\" class=\"header-mark\" aria-label=\"Direct link to section: Indented Code\"\u003e\u003c/a\u003eIndented Code\u003c/h3\u003e\u003cp\u003eOr indent several lines of code by at least four spaces, as in:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-26\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e // Some comments\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e line 1 of code\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e line 2 of code\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e line 3 of code\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\n// Some comments\nline 1 of code\nline 2 of code\nline 3 of code\nThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-27\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003epre\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ecode\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e // Some comments\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e line 1 of code\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e line 2 of code\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e line 3 of code\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ecode\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003epre\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"block-fenced-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#block-fenced-code\" class=\"header-mark\" aria-label=\"Direct link to section: Block Fenced Code\"\u003e\u003c/a\u003eBlock Fenced Code\u003c/h3\u003e\u003cp\u003eUse \u0026ldquo;fences\u0026rdquo; \u003ccode\u003e```\u003c/code\u003e to block in multiple lines of code with a language attribute.\n\u003cdiv class=\"highlight\"\u003e\u003cdiv class=\"chroma\"\u003e\n\u003ctable class=\"lntable\"\u003e\u003ctr\u003e\u003ctd class=\"lntd\"\u003e\n\u003cpre tabindex=\"0\" class=\"chroma\"\u003e\u003ccode\u003e\u003cspan class=\"lnt\"\u003e1\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e2\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e3\n\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\u003c/td\u003e\n\u003ctd class=\"lntd\"\u003e\n\u003cpre tabindex=\"0\" class=\"chroma\"\u003e\u003ccode class=\"language-markdown\" data-lang=\"markdown\"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e```markdown\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e\u003c/span\u003eSample text here...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e```\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e\nThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-28\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003epre\u003c/span\u003e \u003cspan class=\"na\"\u003elanguage-html\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ecode\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eSample text here...\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ecode\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003epre\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"syntax-highlighting\" class=\"headerLink\"\u003e\n \u003ca href=\"#syntax-highlighting\" class=\"header-mark\" aria-label=\"Direct link to section: Syntax Highlighting\"\u003e\u003c/a\u003eSyntax Highlighting\u003c/h3\u003e\u003cp\u003e[GFM]^(GitHub Flavored Markdown) also supports syntax highlighting.\nTo activate it, simply add the file extension of the language you want to use directly after the first code \u0026ldquo;fence\u0026rdquo;, ```js, and syntax highlighting will automatically be applied in the rendered HTML.\nFor example, to apply syntax highlighting to JavaScript code:\n\u003cdiv class=\"highlight\"\u003e\u003cdiv class=\"chroma\"\u003e\n\u003ctable class=\"lntable\"\u003e\u003ctr\u003e\u003ctd class=\"lntd\"\u003e\n\u003cpre tabindex=\"0\" class=\"chroma\"\u003e\u003ccode\u003e\u003cspan class=\"lnt\"\u003e 1\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 2\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 3\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 4\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 5\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 6\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 7\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 8\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e 9\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e10\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e11\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e12\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e13\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e14\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e15\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e16\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e17\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e18\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e19\n\u003c/span\u003e\u003cspan class=\"lnt\"\u003e20\n\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\u003c/td\u003e\n\u003ctd class=\"lntd\"\u003e\n\u003cpre tabindex=\"0\" class=\"chroma\"\u003e\u003ccode class=\"language-markdown\" data-lang=\"markdown\"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e```js\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e\u003c/span\u003e\u003cspan class=\"nx\"\u003egrunt\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"nx\"\u003einitConfig\u003c/span\u003e\u003cspan class=\"p\"\u003e({\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eassemble\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eoptions\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eassets\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;docs/assets\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003edata\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;src/data/*.{json,yml}\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003ehelpers\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;src/custom-helpers.js\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003epartials\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;src/partials/**/*.{hbs,md}\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003epages\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eoptions\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003elayout\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;default.hbs\u0026#39;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003efiles\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"s1\"\u003e\u0026#39;./\u0026#39;\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;src/templates/pages/index.hbs\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e};\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e```\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e\nThe rendered output looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ejs\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-29\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nx\"\u003egrunt\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"nx\"\u003einitConfig\u003c/span\u003e\u003cspan class=\"p\"\u003e({\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eassemble\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eoptions\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eassets\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;docs/assets\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003edata\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;src/data/*.{json,yml}\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003ehelpers\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;src/custom-helpers.js\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003epartials\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;src/partials/**/*.{hbs,md}\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003epages\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eoptions\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003elayout\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s1\"\u003e\u0026#39;default.hbs\u0026#39;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003efiles\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"s1\"\u003e\u0026#39;./\u0026#39;\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;src/templates/pages/index.hbs\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e};\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003ca href=\"https://gohugo.io/content-management/syntax-highlighting/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eSyntax highlighting page\u003c/a\u003e in \u003cstrong\u003eHugo\u003c/strong\u003e Docs introduces more about syntax highlighting, including highlight shortcodes.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch2 id=\"tables\" class=\"headerLink\"\u003e\n \u003ca href=\"#tables\" class=\"header-mark\" aria-label=\"Direct link to section: Tables\"\u003e\u003c/a\u003eTables\u003c/h2\u003e\u003cp\u003eTables are created by adding pipes as dividers between each cell, and by adding a line of dashes (also separated by bars) beneath the header. 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-30\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| Option | Description |\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| ------ | ----------- |\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| data | path to data files to supply the data that will be passed into templates. |\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| engine | engine to be used for processing templates. Handlebars is the default. |\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| ext | extension to be used for dest files. |\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eOption\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eDescription\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003edata\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003epath to data files to supply the data that will be passed into templates.\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eengine\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eengine to be used for processing templates. Handlebars is the default.\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eext\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eextension to be used for dest files.\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eThe HTML looks like this:\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-31\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etable\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ethead\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eth\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eOption\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eth\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003eth\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eDescription\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003eth\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ethead\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etbody\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003edata\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003epath to data files to supply the data that will be passed into templates.\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eengine\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eengine to be used for processing templates. Handlebars is the default.\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eext\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eextension to be used for dest files.\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etd\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etr\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etbody\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003etable\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eRight or center aligned text\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003e\u003cp\u003eAdding a colon on the right side of the dashes below any heading will right align text for that column.\nAdding colons on both sides of the dashes below any heading will center align text for that column.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-47\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| Option | Description |\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e|:------:| -----------:|\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| data | path to data files to supply the data that will be passed into templates. |\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| engine | engine to be used for processing templates. Handlebars is the default. |\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e| ext | extension to be used for dest files. |\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: center\"\u003eOption\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eDescription\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: center\"\u003edata\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003epath to data files to supply the data that will be passed into templates.\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: center\"\u003eengine\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003eengine to be used for processing templates. Handlebars is the default.\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: center\"\u003eext\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003eextension to be used for dest files.\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch2 id=\"links\" class=\"headerLink\"\u003e\n \u003ca href=\"#links\" class=\"header-mark\" aria-label=\"Direct link to section: Links\"\u003e\u003c/a\u003eLinks\u003c/h2\u003e\u003ch3 id=\"basic-link\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-link\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Link\"\u003e\u003c/a\u003eBasic Link\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-32\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ehttps:\u003c/span\u003e\u003cspan class=\"err\"\u003e//\u003c/span\u003e\u003cspan class=\"na\"\u003eassemble\u003c/span\u003e\u003cspan class=\"err\"\u003e.\u003c/span\u003e\u003cspan class=\"na\"\u003eio\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003econtact\u003c/span\u003e\u003cspan class=\"ni\"\u003e@revolunet\u003c/span\u003e\u003cspan class=\"nt\"\u003e.com\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e[\u003cspan class=\"nt\"\u003eAssemble\u003c/span\u003e](\u003cspan class=\"na\"\u003ehttps://assemble.io\u003c/span\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this (hover over the link, there is no tooltip):\n\u003ca href=\"https://assemble.io\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://assemble.io\u003c/a\u003e\n\u003ca href=\"mailto:contact@revolunet.com\" rel=\"\"\u003econtact@revolunet.com\u003c/a\u003e\n\u003ca href=\"https://assemble.io\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAssemble\u003c/a\u003e\nThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-33\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e \u003cspan class=\"na\"\u003ehref\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;https://assemble.io\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003ehttps://assemble.io\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e \u003cspan class=\"na\"\u003ehref\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;mailto:contact@revolunet.com\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003econtact@revolunet.com\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e \u003cspan class=\"na\"\u003ehref\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;https://assemble.io\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eAssemble\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"add-a-title\" class=\"headerLink\"\u003e\n \u003ca href=\"#add-a-title\" class=\"header-mark\" aria-label=\"Direct link to section: Add a Title\"\u003e\u003c/a\u003eAdd a Title\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-34\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e[\u003cspan class=\"nt\"\u003eUpstage\u003c/span\u003e](\u003cspan class=\"na\"\u003ehttps://github.com/upstage/ \u0026#34;Visit Upstage!\u0026#34;\u003c/span\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe rendered output looks like this (hover over the link, there is a tooltip):\n\u003ca href=\"https://github.com/upstage/\" title=\"Visit Upstage!\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eUpstage\u003c/a\u003e\nThe HTML looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-35\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e \u003cspan class=\"na\"\u003ehref\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;https://github.com/upstage/\u0026#34;\u003c/span\u003e \u003cspan class=\"na\"\u003etitle\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;Visit Upstage!\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003eUpstage\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003ea\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"named-anchors\" class=\"headerLink\"\u003e\n \u003ca href=\"#named-anchors\" class=\"header-mark\" aria-label=\"Direct link to section: Named Anchors\"\u003e\u003c/a\u003eNamed Anchors\u003c/h3\u003e\u003cp\u003eNamed anchors enable you to jump to the specified anchor point on the same page. For example, each of these chapters:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-36\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e## Table of Contents\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e*\u003c/span\u003e [\u003cspan class=\"nt\"\u003eChapter 1\u003c/span\u003e](\u003cspan class=\"na\"\u003e#chapter-1\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e*\u003c/span\u003e [\u003cspan class=\"nt\"\u003eChapter 2\u003c/span\u003e](\u003cspan class=\"na\"\u003e#chapter-2\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e * [\u003cspan class=\"nt\"\u003eChapter 3\u003c/span\u003e](\u003cspan class=\"na\"\u003e#chapter-3\u003c/span\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003ewill jump to these sections:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-37\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e## Chapter 1 \u0026lt;a id=\u0026#34;chapter-1\u0026#34;\u0026gt;\u0026lt;/a\u0026gt;\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e\u003c/span\u003eContent for chapter one.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e## Chapter 2 \u0026lt;a id=\u0026#34;chapter-2\u0026#34;\u0026gt;\u0026lt;/a\u0026gt;\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e\u003c/span\u003eContent for chapter two.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e## Chapter 3 \u0026lt;a id=\u0026#34;chapter-3\u0026#34;\u0026gt;\u0026lt;/a\u0026gt;\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"gu\"\u003e\u003c/span\u003eContent for chapter three.\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eThe specific placement of the anchor tag seems to be arbitrary. They are placed inline here since it seems to be unobtrusive, and it works.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch2 id=\"footnotes\" class=\"headerLink\"\u003e\n \u003ca href=\"#footnotes\" class=\"header-mark\" aria-label=\"Direct link to section: Footnotes\"\u003e\u003c/a\u003eFootnotes\u003c/h2\u003e\u003cp\u003eFootnotes allow you to add notes and references without cluttering the body of the document. When you create a footnote, a superscript number with a link appears where you added the footnote reference. Readers can click the link to jump to the content of the footnote at the bottom of the page.\nTo create a footnote reference, add a caret and an identifier inside brackets (\u003csup id=\"fnref:1\"\u003e\u003ca href=\"#fn:1\" class=\"footnote-ref\" role=\"doc-noteref\"\u003e1\u003c/a\u003e\u003c/sup\u003e). Identifiers can be numbers or words, but they can\u0026rsquo;t contain spaces or tabs. Identifiers only correlate the footnote reference with the footnote itself — in the output, footnotes are numbered sequentially.\nAdd the footnote using another caret and number inside brackets with a colon and text (\u003csup id=\"fnref1:1\"\u003e\u003ca href=\"#fn:1\" class=\"footnote-ref\" role=\"doc-noteref\"\u003e1\u003c/a\u003e\u003c/sup\u003e: My footnote.). You don\u0026rsquo;t have to put footnotes at the end of the document. You can put them anywhere except inside other elements like lists, block quotes, and tables.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-38\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis is a digital footnote[^1]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis is a footnote with \u0026#34;label\u0026#34;[^label]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e[^1]: This is a digital footnote\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e[^label]: This is a footnote with \u0026#34;label\u0026#34;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis is a digital footnote\u003csup id=\"fnref2:1\"\u003e\u003ca href=\"#fn:1\" class=\"footnote-ref\" role=\"doc-noteref\"\u003e1\u003c/a\u003e\u003c/sup\u003e\nThis is a footnote with \u0026ldquo;label\u0026rdquo;\u003csup id=\"fnref:2\"\u003e\u003ca href=\"#fn:2\" class=\"footnote-ref\" role=\"doc-noteref\"\u003e2\u003c/a\u003e\u003c/sup\u003e\u003c/p\u003e\n\u003ch2 id=\"images\" class=\"headerLink\"\u003e\n \u003ca href=\"#images\" class=\"header-mark\" aria-label=\"Direct link to section: Images\"\u003e\u003c/a\u003eImages\u003c/h2\u003e\u003cp\u003eImages have a similar syntax to links but include a preceding exclamation point.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-39\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e![\u003cspan class=\"nt\"\u003eMinion\u003c/span\u003e](\u003cspan class=\"na\"\u003ehttps://octodex.github.com/images/minion.png\u003c/span\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://octodex.github.com/images/minion.png' alt=\"Minion\" \u003e\nor:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-40\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e![\u003cspan class=\"nt\"\u003eAlt text\u003c/span\u003e](\u003cspan class=\"na\"\u003ehttps://octodex.github.com/images/stormtroopocat.jpg \u0026#34;The Stormtroopocat\u0026#34;\u003c/span\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://octodex.github.com/images/stormtroopocat.jpg\" title=\"Alt text\" data-thumbnail=\"https://octodex.github.com/images/stormtroopocat.jpg\" data-sub-html=\"\u003ch2\u003eThe Stormtroopocat\u003c/h2\u003e\u003cp\u003eAlt text\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://octodex.github.com/images/stormtroopocat.jpg' alt=\"Alt text\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eThe Stormtroopocat\u003c/figcaption\u003e\n\u003c/figure\u003e\nLike links, images also have a footnote style syntax:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-41\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e![Alt text][id]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://octodex.github.com/images/dojocat.jpg\" title=\"Alt text\" data-thumbnail=\"https://octodex.github.com/images/dojocat.jpg\" data-sub-html=\"\u003ch2\u003eThe Dojocat\u003c/h2\u003e\u003cp\u003eAlt text\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://octodex.github.com/images/dojocat.jpg' alt=\"Alt text\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eThe Dojocat\u003c/figcaption\u003e\n\u003c/figure\u003e\nWith a reference later in the document defining the URL location:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003emarkdown\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-42\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e[id]: https://octodex.github.com/images/dojocat.jpg \u0026#34;The Dojocat\u0026#34;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThat\u0026rsquo;s the article about basic Markdown syntax. Hope it\u0026rsquo;s useful. 😙\u003c/p\u003e\n\u003cdiv class=\"footnotes\" role=\"doc-endnotes\"\u003e\n\u003chr\u003e\n\u003col\u003e\n\u003cli id=\"fn:1\"\u003e\n\u003cp\u003eThis is a digital footnote\u0026#160;\u003ca href=\"#fnref:1\" class=\"footnote-backref\" role=\"doc-backlink\"\u003e\u0026#x21a9;\u0026#xfe0e;\u003c/a\u003e\u0026#160;\u003ca href=\"#fnref1:1\" class=\"footnote-backref\" role=\"doc-backlink\"\u003e\u0026#x21a9;\u0026#xfe0e;\u003c/a\u003e\u0026#160;\u003ca href=\"#fnref2:1\" class=\"footnote-backref\" role=\"doc-backlink\"\u003e\u0026#x21a9;\u0026#xfe0e;\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli id=\"fn:2\"\u003e\n\u003cp\u003eThis is a footnote with \u0026ldquo;label\u0026rdquo;\u0026#160;\u003ca href=\"#fnref:2\" class=\"footnote-backref\" role=\"doc-backlink\"\u003e\u0026#x21a9;\u0026#xfe0e;\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003c/div\u003e", "language": "en" }, { "title" : "A Brief History of GNU/Linux", "date_published" : "2025-09-28T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/gnu-linux/", "url" : "https://www.aantriono.com/en/gnu-linux/", "summary": "This article discusses the brief history of the GNU/Linux operating system. For more complete sources, please read About GNU. For reliable and complete sources about Linux, please read About Linux.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the brief history of the GNU/Linux operating system.\nFor more complete sources, please read \u003ca href=\"https://www.gnu.org/gnu/gnu.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAbout GNU\u003c/a\u003e.\nFor reliable and complete sources about Linux, please read \u003ca href=\"https://www.linux.org/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAbout Linux\u003c/a\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"understanding-gnulinux-a-free-and-open-operating-system\" class=\"headerLink\"\u003e\n \u003ca href=\"#understanding-gnulinux-a-free-and-open-operating-system\" class=\"header-mark\" aria-label=\"Direct link to section: Understanding GNU/Linux: A Free and Open Operating System\"\u003e\u003c/a\u003eUnderstanding GNU/Linux: A Free and Open Operating System\u003c/h2\u003e\u003cp\u003eGNU/Linux is an operating system consisting of two main components: the GNU project and the Linux kernel. This operating system is known for its free and open nature, which allows users to modify, distribute, and use it without the strict restrictions found in proprietary operating systems. GNU/Linux has become the primary choice for many users, from individuals to large organizations, due to its flexibility, security, and stability.\u003c/p\u003e\n\u003ch3 id=\"a-brief-history-of-gnulinux\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-brief-history-of-gnulinux\" class=\"header-mark\" aria-label=\"Direct link to section: A Brief History of GNU/Linux\"\u003e\u003c/a\u003eA Brief History of GNU/Linux\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eThe GNU Project\u003c/strong\u003e:\nThe GNU project was started in 1983 by Richard Stallman, a programmer and free software activist. The main goal of this project was to create a completely free and open operating system. GNU stands for \u0026ldquo;GNU\u0026rsquo;s Not Unix,\u0026rdquo; indicating that the system is inspired by Unix but does not use Unix source code. The project successfully developed many important components such as a compiler (GCC), text editor (Emacs), and other system utilities. However, the GNU project still lacked a functional kernel.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe Linux Kernel\u003c/strong\u003e:\nIn 1991, Linus Torvalds, a student from Finland, developed the Linux kernel as a personal project. This kernel was released under a free license, and over time, many developers contributed to perfecting it. The Linux kernel was then combined with GNU components to create a complete operating system known as GNU/Linux.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"what-is-gnulinux\" class=\"headerLink\"\u003e\n \u003ca href=\"#what-is-gnulinux\" class=\"header-mark\" aria-label=\"Direct link to section: What is GNU/Linux?\"\u003e\u003c/a\u003eWhat is GNU/Linux?\u003c/h3\u003e\u003cp\u003eGNU/Linux is an operating system that combines the Linux kernel with software from the GNU project. This operating system is released under the GPL (General Public License), which ensures that users have the freedom to use, study, modify, and distribute its source code. Some main characteristics of GNU/Linux include:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eFree and Open\u003c/strong\u003e:\nGNU/Linux is open source software, meaning its source code is available to the public. Users can modify and redistribute this operating system according to their needs.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eStability and Security\u003c/strong\u003e:\nGNU/Linux is known for its high stability and security. The system rarely experiences crashes or virus attacks, making it a popular choice for servers and critical systems.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFlexibility\u003c/strong\u003e:\nGNU/Linux can run on various hardware, from personal computers, servers, to embedded devices such as routers and smartphones. Additionally, there are many Linux distributions (distros) that offer various configurations and user interfaces.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eStrong Community\u003c/strong\u003e:\nGNU/Linux is supported by a global community consisting of developers, users, and organizations. This community actively contributes to development, documentation, and technical support.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"gnulinux-distributions\" class=\"headerLink\"\u003e\n \u003ca href=\"#gnulinux-distributions\" class=\"header-mark\" aria-label=\"Direct link to section: GNU/Linux Distributions\"\u003e\u003c/a\u003eGNU/Linux Distributions\u003c/h3\u003e\u003cp\u003eGNU/Linux distributions (distros) are versions of the operating system packaged with additional software, such as package managers, user interfaces, and applications. Some popular distros include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eUbuntu\u003c/strong\u003e: A user-friendly distro suitable for beginners.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDebian\u003c/strong\u003e: A distro known for its stability and commitment to free software principles.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFedora\u003c/strong\u003e: A distro supported by Red Hat and focused on innovation.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eArch Linux\u003c/strong\u003e: A distro aimed at advanced users and offering complete freedom in customization.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"advantages-of-gnulinux\" class=\"headerLink\"\u003e\n \u003ca href=\"#advantages-of-gnulinux\" class=\"header-mark\" aria-label=\"Direct link to section: Advantages of GNU/Linux\"\u003e\u003c/a\u003eAdvantages of GNU/Linux\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eNo License Costs\u003c/strong\u003e:\nGNU/Linux can be downloaded and used for free, saving on license costs typically incurred for proprietary operating systems.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eHigh Customization\u003c/strong\u003e:\nUsers can customize the system according to their needs, from the user interface to core system components.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eWide Hardware Support\u003c/strong\u003e:\nGNU/Linux supports various types of hardware, including older devices that may no longer be supported by other operating systems.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eRich Software Ecosystem\u003c/strong\u003e:\nThere are thousands of open source applications available for GNU/Linux, covering various needs such as office work, multimedia, and software development.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eGNU/Linux is a powerful, flexible, and free operating system that has changed the way the world views software. With the combination of the GNU project and the Linux kernel, this system offers a stable, secure, and customizable solution for various needs. Whether you are a regular user, developer, or system administrator, GNU/Linux provides a reliable and efficient platform for working and innovating.\u003c/p\u003e", "language": "en" }, { "title" : "Differences Between BSD and Linux", "date_published" : "2025-10-17T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/difference-between-bsd-and-linux/", "url" : "https://www.aantriono.com/en/difference-between-bsd-and-linux/", "summary": "This article discusses the differences between BSD and GNU/Linux operating systems.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article discusses the differences between BSD and GNU/Linux operating systems.\u003c/p\u003e\n\u003chr\u003e\n\u003cdiv class=\"details admonition note open\"\u003e\n \u003cdiv class=\"details-summary admonition-title\"\u003e\n \u003cspan class=\"icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M497.9 142.1l-46.1 46.1c-4.7 4.7-12.3 4.7-17 0l-111-111c-4.7-4.7-4.7-12.3 0-17l46.1-46.1c18.7-18.7 49.1-18.7 67.9 0l60.1 60.1c18.8 18.7 18.8 49.1 0 67.9zM284.2 99.8L21.6 362.4.4 483.9c-2.9 16.4 11.4 30.6 27.8 27.8l121.5-21.3 262.6-262.6c4.7-4.7 4.7-12.3 0-17l-111-111c-4.8-4.7-12.4-4.7-17.1 0zM124.1 339.9c-5.5-5.5-5.5-14.3 0-19.8l154-154c5.5-5.5 14.3-5.5 19.8 0s5.5 14.3 0 19.8l-154 154c-5.5 5.5-14.3 5.5-19.8 0zM88 424h48v36.3l-64.5 11.3-31.1-31.1L51.7 376H88v48z\"/\u003e\u003c/svg\u003e\u003c/span\u003eNote\u003cspan class=\"details-icon\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 256 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/div\u003e\n \u003cdiv class=\"details-content\"\u003e\n \u003cdiv class=\"admonition-content\"\u003eWhen it comes to the most influential operating systems of all time, you can\u0026rsquo;t go without mentioning Unix, Linux, and BSD. Our modern world couldn\u0026rsquo;t exist without these operating systems.\nWhile they share similarities in their basic principles and functions, there are also distinct differences that set them apart.\nThat said, in this article, our goal is to explain and compare these operating systems, exploring their origins, characteristics, and uses.\nLet\u0026rsquo;s dive in.\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch1 id=\"most-influential-operating-systems-unix-bsd-and-linux\" class=\"headerLink\"\u003e\n \u003ca href=\"#most-influential-operating-systems-unix-bsd-and-linux\" class=\"header-mark\" aria-label=\"Direct link to section: Most Influential Operating Systems: Unix, BSD, and Linux\"\u003e\u003c/a\u003eMost Influential Operating Systems: Unix, BSD, and Linux\u003c/h1\u003e\u003ch2 id=\"this-article-will-explain-the-origins-characteristics-and-uses-of-unix-bsd-and-linux-while-comparing-their-roles-in-the-modern-computing-landscape\" class=\"headerLink\"\u003e\n \u003ca href=\"#this-article-will-explain-the-origins-characteristics-and-uses-of-unix-bsd-and-linux-while-comparing-their-roles-in-the-modern-computing-landscape\" class=\"header-mark\" aria-label=\"Direct link to section: When discussing the most influential operating systems of all time, it\u0026rsquo;s impossible to overlook Unix, BSD, and Linux. Our modern world, from the internet to mobile devices, stands on the foundation they built. All three share basic principles—multi-user, multitasking, modularity—yet evolved with unique characteristics that distinguish them from one another. This article will explain the origins, characteristics, and uses of Unix, BSD, and Linux, while comparing their roles in the modern computing landscape.\"\u003e\u003c/a\u003eWhen discussing the most influential operating systems of all time, it\u0026rsquo;s impossible to overlook \u003cstrong\u003eUnix, BSD, and Linux\u003c/strong\u003e. Our modern world, from the internet to mobile devices, stands on the foundation they built. All three share basic principles—multi-user, multitasking, modularity—yet evolved with unique characteristics that distinguish them from one another.\nThis article will explain the origins, characteristics, and uses of Unix, BSD, and Linux, while comparing their roles in the modern computing landscape.\u003c/h2\u003e\u003ch2 id=\"unix-the-foundation-of-modern-operating-systems\" class=\"headerLink\"\u003e\n \u003ca href=\"#unix-the-foundation-of-modern-operating-systems\" class=\"header-mark\" aria-label=\"Direct link to section: Unix: The Foundation of Modern Operating Systems\"\u003e\u003c/a\u003e\u003cstrong\u003eUnix: The Foundation of Modern Operating Systems\u003c/strong\u003e\u003c/h2\u003e\u003ch2 id=\"additionally-unix-gave-birth-to-the-posix-portable-operating-system-interface-standard-which-ensures-compatibility-among-modern-operating-systems\" class=\"headerLink\"\u003e\n \u003ca href=\"#additionally-unix-gave-birth-to-the-posix-portable-operating-system-interface-standard-which-ensures-compatibility-among-modern-operating-systems\" class=\"header-mark\" aria-label=\"Direct link to section: Unix was born in the late 1960s at AT\u0026amp;T Bell Labs, led by Ken Thompson and Dennis Ritchie. They created a simple yet powerful multi-user and multitasking operating system. Unix\u0026rsquo;s design principles—simplicity, modularity, hierarchical file system—became the foundation for many subsequent OS developments. One of its key innovations was the \u0026ldquo;pipe\u0026rdquo;, which allowed the output of one program to be used as input for another, enhancing command-line flexibility. Unix evolved rapidly during the 1970s-1980s, adopted by universities, research institutions, and corporations. Its portability (thanks to being rewritten in C language) made it an industry standard. From Unix came many commercial variants (Solaris, HP-UX, AIX) as well as inspiration for open-source systems like BSD and Linux. Additionally, Unix gave birth to the POSIX (Portable Operating System Interface) standard, which ensures compatibility among modern operating systems.\"\u003e\u003c/a\u003eUnix was born in the late 1960s at \u003cstrong\u003eAT\u0026amp;T Bell Labs\u003c/strong\u003e, led by \u003cstrong\u003eKen Thompson\u003c/strong\u003e and \u003cstrong\u003eDennis Ritchie\u003c/strong\u003e. They created a simple yet powerful \u003cstrong\u003emulti-user and multitasking\u003c/strong\u003e operating system.\nUnix\u0026rsquo;s design principles—\u003cstrong\u003esimplicity, modularity, hierarchical file system\u003c/strong\u003e—became the foundation for many subsequent OS developments. One of its key innovations was the \u003cstrong\u003e\u0026ldquo;pipe\u0026rdquo;\u003c/strong\u003e, which allowed the output of one program to be used as input for another, enhancing command-line flexibility.\nUnix evolved rapidly during the 1970s-1980s, adopted by universities, research institutions, and corporations. Its portability (thanks to being rewritten in C language) made it an industry standard. From Unix came many commercial variants (Solaris, HP-UX, AIX) as well as inspiration for open-source systems like BSD and Linux.\nAdditionally, Unix gave birth to the \u003cstrong\u003ePOSIX (Portable Operating System Interface)\u003c/strong\u003e standard, which ensures compatibility among modern operating systems.\u003c/h2\u003e\u003ch2 id=\"bsd-innovation-from-berkeley\" class=\"headerLink\"\u003e\n \u003ca href=\"#bsd-innovation-from-berkeley\" class=\"header-mark\" aria-label=\"Direct link to section: BSD: Innovation from Berkeley\"\u003e\u003c/a\u003e\u003cstrong\u003eBSD: Innovation from Berkeley\u003c/strong\u003e\u003c/h2\u003e\u003cp\u003e\u003cstrong\u003eBSD (Berkeley Software Distribution)\u003c/strong\u003e emerged in the late 1970s at the University of California, Berkeley, as an enhancement to the Unix source code. BSD contributed important innovations, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eTCP/IP stack\u003c/strong\u003e, the main foundation of the modern internet\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003evi editor\u003c/strong\u003e, which remains popular to this day\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"modern-bsd-variants\" class=\"headerLink\"\u003e\n \u003ca href=\"#modern-bsd-variants\" class=\"header-mark\" aria-label=\"Direct link to section: Modern BSD Variants\"\u003e\u003c/a\u003e\u003cstrong\u003eModern BSD Variants\u003c/strong\u003e\u003c/h3\u003e\u003cp\u003eBSD evolved into an independent ecosystem, with main derivatives:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eFreeBSD\u003c/strong\u003e – focuses on performance \u0026amp; scalability, popular for servers and embedded systems\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eOpenBSD\u003c/strong\u003e – emphasizes \u003cstrong\u003esecurity \u0026amp; code auditing\u003c/strong\u003e, widely used for firewalls and network infrastructure\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNetBSD\u003c/strong\u003e – known for portability, can run on almost any hardware architecture\nBSD also became the basis for major technologies: \u003cstrong\u003eDarwin (macOS \u0026amp; iOS kernel)\u003c/strong\u003e, and the \u003cstrong\u003ePlayStation 4\u003c/strong\u003e operating system.\nOne thing that distinguishes BSD from Linux is its licensing. The \u003cstrong\u003eBSD License\u003c/strong\u003e is more permissive compared to \u003cstrong\u003eLinux\u0026rsquo;s GPL\u003c/strong\u003e, allowing companies to adopt BSD code without having to open their modifications—the reason Apple chose BSD as the basis for macOS.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"linux-open-source-for-everyone\" class=\"headerLink\"\u003e\n \u003ca href=\"#linux-open-source-for-everyone\" class=\"header-mark\" aria-label=\"Direct link to section: Linux: Open Source for Everyone\"\u003e\u003c/a\u003e\u003cstrong\u003eLinux: Open Source for Everyone\u003c/strong\u003e\u003c/h2\u003e\u003cp\u003eLinux emerged in 1991, created by \u003cstrong\u003eLinus Torvalds\u003c/strong\u003e at the University of Helsinki. Unlike BSD, Linux \u003cstrong\u003ewas not directly derived from Unix code\u003c/strong\u003e, but was written from scratch with Unix philosophy as inspiration.\nLinux is compatible with Unix standards (POSIX), modular, and \u003cstrong\u003eopen-source\u003c/strong\u003e under the \u003cstrong\u003eGNU General Public License (GPL)\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"integration-with-gnu\" class=\"headerLink\"\u003e\n \u003ca href=\"#integration-with-gnu\" class=\"header-mark\" aria-label=\"Direct link to section: Integration with GNU\"\u003e\u003c/a\u003e\u003cstrong\u003eIntegration with GNU\u003c/strong\u003e\u003c/h3\u003e\u003cp\u003eBefore Linux was born, the \u003cstrong\u003eGNU Project\u003c/strong\u003e (Richard Stallman, 1983) had already provided many important components:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eGNU Compiler Collection (GCC)\u003c/li\u003e\n\u003cli\u003eGNU C Library (glibc)\u003c/li\u003e\n\u003cli\u003eGNU Core Utilities\nHowever, GNU lacked a fully functional kernel. The Linux kernel filled this gap, so the combination of \u003cstrong\u003eGNU/Linux\u003c/strong\u003e resulted in a complete and free operating system.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"linux-dominance\" class=\"headerLink\"\u003e\n \u003ca href=\"#linux-dominance\" class=\"header-mark\" aria-label=\"Direct link to section: Linux Dominance\"\u003e\u003c/a\u003e\u003cstrong\u003eLinux Dominance\u003c/strong\u003e\u003c/h3\u003e\u003cp\u003eToday, Linux:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ePowers \u003cstrong\u003e\u0026gt;95% of the world\u0026rsquo;s supercomputers\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eForms the core of \u003cstrong\u003eAndroid\u003c/strong\u003e, the most popular smartphone operating system\u003c/li\u003e\n\u003cli\u003eRuns the majority of \u003cstrong\u003eweb servers \u0026amp; cloud computing\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eUsed in IoT, embedded devices, and desktops\nPopular distributions include \u003cstrong\u003eUbuntu, Debian, Fedora, Arch Linux, and Red Hat Enterprise Linux\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"kernel-and-architecture\" class=\"headerLink\"\u003e\n \u003ca href=\"#kernel-and-architecture\" class=\"header-mark\" aria-label=\"Direct link to section: Kernel and Architecture\"\u003e\u003c/a\u003e\u003cstrong\u003eKernel and Architecture\u003c/strong\u003e\u003c/h2\u003e\u003cp\u003eAll three systems use \u003cstrong\u003emonolithic kernels\u003c/strong\u003e, but with different approaches:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eUnix\u003c/strong\u003e: large kernel managing all services \u0026amp; hardware\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBSD\u003c/strong\u003e: monolithic kernel with enhancements like \u003cstrong\u003eZFS filesystem\u003c/strong\u003e, \u003cstrong\u003eoptimized network stack\u003c/strong\u003e, and security features like \u003cstrong\u003eMandatory Access Control (MAC)\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLinux\u003c/strong\u003e: \u003cstrong\u003emodular\u003c/strong\u003e monolithic kernel, allowing drivers \u0026amp; features to be loaded/unloaded as needed. This makes Linux flexible for various devices, from IoT to supercomputers\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/difference-between-bsd-and-linux/Unix.webp' srcset=\"/difference-between-bsd-and-linux/Unix_hu_63409e7d50b8945.webp 800w, /difference-between-bsd-and-linux/Unix_hu_252477e66a04b1f0.webp 1200w, /difference-between-bsd-and-linux/Unix_hu_4cf0a17b94abef9d.webp 1600w\" alt=\"OS Development\" height=\"665\" width=\"952\"\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"ecosystem-and-community\" class=\"headerLink\"\u003e\n \u003ca href=\"#ecosystem-and-community\" class=\"header-mark\" aria-label=\"Direct link to section: Ecosystem and Community\"\u003e\u003c/a\u003e\u003cstrong\u003eEcosystem and Community\u003c/strong\u003e\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eUnix\u003c/strong\u003e: now mostly survives in commercial variants (Solaris, AIX) in companies with legacy systems\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBSD\u003c/strong\u003e: small but dedicated community, known for code quality and documentation\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLinux\u003c/strong\u003e: has the largest ecosystem with thousands of distributions and active global communities\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"usage-and-considerations\" class=\"headerLink\"\u003e\n \u003ca href=\"#usage-and-considerations\" class=\"header-mark\" aria-label=\"Direct link to section: Usage and Considerations\"\u003e\u003c/a\u003e\u003cstrong\u003eUsage and Considerations\u003c/strong\u003e\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eUnix\u003c/strong\u003e → Still used in large corporations with legacy systems (banking, telecommunications, aerospace)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBSD\u003c/strong\u003e → Suitable for firewalls, routers, and systems with high security/stability demands\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLinux\u003c/strong\u003e → Most flexible, used in cloud, servers, desktops, IoT, and supercomputers\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"interoperability-and-compatibility\" class=\"headerLink\"\u003e\n \u003ca href=\"#interoperability-and-compatibility\" class=\"header-mark\" aria-label=\"Direct link to section: Interoperability and Compatibility\"\u003e\u003c/a\u003e\u003cstrong\u003eInteroperability and Compatibility\u003c/strong\u003e\u003c/h2\u003e\u003ch2 id=\"all-three-share-the-same-heritage-so-many-applications-can-be-easily-ported-between-systems-especially-those-written-in-cc-core-command-line-utilities-like-ls-grep-awk-are-also-similar-making-it-relatively-easy-for-users-of-one-system-to-adapt-to-others\" class=\"headerLink\"\u003e\n \u003ca href=\"#all-three-share-the-same-heritage-so-many-applications-can-be-easily-ported-between-systems-especially-those-written-in-cc-core-command-line-utilities-like-ls-grep-awk-are-also-similar-making-it-relatively-easy-for-users-of-one-system-to-adapt-to-others\" class=\"header-mark\" aria-label=\"Direct link to section: All three share the same heritage, so many applications can be easily ported between systems, especially those written in C/C\u0026#43;\u0026#43;. Core command-line utilities (like ls, grep, awk) are also similar, making it relatively easy for users of one system to adapt to others.\"\u003e\u003c/a\u003eAll three share the same heritage, so \u003cstrong\u003emany applications can be easily ported\u003c/strong\u003e between systems, especially those written in C/C++. Core command-line utilities (like \u003ccode\u003els\u003c/code\u003e, \u003ccode\u003egrep\u003c/code\u003e, \u003ccode\u003eawk\u003c/code\u003e) are also similar, making it relatively easy for users of one system to adapt to others.\u003c/h2\u003e\u003ch2 id=\"comparison-table-unix-bsd-and-linux\" class=\"headerLink\"\u003e\n \u003ca href=\"#comparison-table-unix-bsd-and-linux\" class=\"header-mark\" aria-label=\"Direct link to section: Comparison Table: Unix, BSD, and Linux\"\u003e\u003c/a\u003e\u003cstrong\u003eComparison Table: Unix, BSD, and Linux\u003c/strong\u003e\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eAspect\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eUnix\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eBSD\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eLinux\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eOrigins\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e1969, AT\u0026amp;T Bell Labs\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e1977, UC Berkeley, Unix derivative\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e1991, Linus Torvalds, independent\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eLicense\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eProprietary (AT\u0026amp;T, commercial)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eBSD License (permissive)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eGPL (copyleft, open-source)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eFocus\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCommercial systems, stability\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNetworking, security, portability\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFlexibility, community, open-source\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eVariants\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eSolaris, AIX, HP-UX\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFreeBSD, OpenBSD, NetBSD\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eUbuntu, Debian, Fedora, RHEL, Arch\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eUsage\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCorporations, mission-critical\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eServers, firewalls, basis for macOS \u0026amp; PS4\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eCloud, servers, desktop, Android, IoT\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eCurrent Status\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eLimited, declining dominance\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNiche but important (security \u0026amp; research)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eGlobal dominance, extensive ecosystem\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003e\u003cstrong\u003eConclusion\u003c/strong\u003e\u003c/h2\u003e\u003cp\u003eUnix, BSD, and Linux are not just operating systems, but the \u003cstrong\u003ebackbone of the modern digital world\u003c/strong\u003e.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eUnix\u003c/strong\u003e laid the foundation with design principles and POSIX standards\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBSD\u003c/strong\u003e contributed important innovations, especially in networking and security\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLinux\u003c/strong\u003e extended influence worldwide through the open-source model, from global servers to smartphones in our pockets\nTogether, these three form an ecosystem that continues to drive the evolution of computing technology.\u003c/li\u003e\n\u003c/ul\u003e", "language": "en" }, { "title" : "Drawing a Rose Curve with TikZ in LaTeX", "date_published" : "2026-08-08T21:00:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/rose-curve/", "url" : "https://www.aantriono.com/en/rose-curve/", "summary": " IntroductionThe rose curve is one of the most visually beautiful mathematical curves. It was first studied in depth by the Italian mathematician Guido Grandi in 1725. Its flower-petal-like shape makes it a favorite in mathematical art and decorative design.\nThis article discusses the mathematical analysis of the rose curve and the LaTeX TikZ coding technique to draw it, complete with source code, compiled output, and experiments with different petal counts.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThe rose curve is one of the most visually beautiful mathematical curves. It was first studied in depth by the Italian mathematician \u003cstrong\u003eGuido Grandi\u003c/strong\u003e in 1725. Its flower-petal-like shape makes it a favorite in mathematical art and decorative design.\u003c/p\u003e\n\u003cp\u003eThis article discusses the mathematical analysis of the rose curve and the LaTeX TikZ coding technique to draw it, complete with source code, compiled output, and experiments with different petal counts.\u003c/p\u003e\n\u003ch2 id=\"mathematical-analysis-the-polar-equation-of-the-rose\" class=\"headerLink\"\u003e\n \u003ca href=\"#mathematical-analysis-the-polar-equation-of-the-rose\" class=\"header-mark\" aria-label=\"Direct link to section: Mathematical Analysis: The Polar Equation of the Rose\"\u003e\u003c/a\u003eMathematical Analysis: The Polar Equation of the Rose\u003c/h2\u003e\u003cp\u003eThe rose curve is defined in \u003cstrong\u003epolar coordinates\u003c/strong\u003e by the general equation:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er = a\\cos(k\\theta) \\quad \\text{or} \\quad r = a\\sin(k\\theta)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003esin\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = amplitude (maximum petal length from the center),\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = constant that determines the number of petals,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = polar angle (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0^\\circ \\leq \\theta \\leq 360^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8101em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8304em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e).\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe number of petals is determined by the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eIf \u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is odd\u003c/strong\u003e → the curve has \u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e petals\u003c/strong\u003e (e.g., \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek = 5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e → 5 petals),\u003c/li\u003e\n\u003cli\u003eIf \u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is even\u003c/strong\u003e → the curve has \u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2k\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e petals\u003c/strong\u003e (e.g., \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek = 8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e → 16 petals).\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIn this article we use \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er = a\\cos(k\\theta)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2.9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea = 2.9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2.9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and vary \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to obtain different rose shapes.\u003c/p\u003e\n\u003ch2 id=\"latex-source-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-source-code\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Source Code\"\u003e\u003c/a\u003eLaTeX Source Code\u003c/h2\u003e\u003cp\u003eHere is the complete TikZ code to draw a rose with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek = 5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz,border=10pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[very thick, red!75!black, domain=0:360, samples=600, variable=\\t]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e plot (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(5*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*cos(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(5*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*sin(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"compiled-output\" class=\"headerLink\"\u003e\n \u003ca href=\"#compiled-output\" class=\"header-mark\" aria-label=\"Direct link to section: Compiled Output\"\u003e\u003c/a\u003eCompiled Output\u003c/h2\u003e\u003cp\u003eThe code above was compiled with the \u003cstrong\u003eTectonic\u003c/strong\u003e engine (XeLaTeX-based), producing the following image:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/rose-curve/rose5.png' srcset=\"/rose-curve/rose5_hu_30af54d0910eca56.webp 800w, /rose-curve/rose5_hu_386da74db8b268e5.webp 1200w, /rose-curve/rose5_hu_20ffb699863e00d0.webp 1600w\" alt=\"Rose curve with 5 petals (k = 5)\" height=\"445\" width=\"427\"\u003e\u003c/p\u003e\n\u003cp\u003eA rose with \u003cstrong\u003e5 petals\u003c/strong\u003e because \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek = 5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (an odd number). The petals are symmetrically arranged around the center, resembling a perfectly bloomed flower.\u003c/p\u003e\n\u003ch2 id=\"explaining-the-code-structure--logic\" class=\"headerLink\"\u003e\n \u003ca href=\"#explaining-the-code-structure--logic\" class=\"header-mark\" aria-label=\"Direct link to section: Explaining the Code Structure \u0026amp; Logic\"\u003e\u003c/a\u003eExplaining the Code Structure \u0026amp; Logic\u003c/h2\u003e\u003cp\u003eLet us break down the code line by line:\u003c/p\u003e\n\u003ch3 id=\"1-the-standalone-document-class\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-the-standalone-document-class\" class=\"header-mark\" aria-label=\"Direct link to section: 1. The Standalone Document Class\"\u003e\u003c/a\u003e1. The Standalone Document Class\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz,border=10pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe \u003ccode\u003estandalone\u003c/code\u003e class crops the output tightly to the drawing (no full page), with a \u003ccode\u003e10pt\u003c/code\u003e margin. The \u003ccode\u003etikz\u003c/code\u003e option loads the core TikZ library automatically.\u003c/p\u003e\n\u003ch3 id=\"2-the-draw-command-with-plot\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-the-draw-command-with-plot\" class=\"header-mark\" aria-label=\"Direct link to section: 2. The \\draw Command with plot\"\u003e\u003c/a\u003e2. The \\draw Command with plot\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[very thick, red!75!black, domain=0:360, samples=600, variable=\\t]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e plot (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(5*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*cos(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(5*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*sin(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe key parts of this command:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eOption\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003ePurpose\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003edomain=0:360\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eThe angle range \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e from 0° to 360° (one full revolution)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003esamples=600\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eNumber of points computed along the curve — the more, the smoother\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003evariable=\\t\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eDefines the angle variable \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e as \u003ccode\u003e\\t\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003eplot ({...}, {...})\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eComputes Cartesian coordinates from polar coordinates\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"3-converting-polar-to-cartesian\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-converting-polar-to-cartesian\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Converting Polar to Cartesian\"\u003e\u003c/a\u003e3. Converting Polar to Cartesian\u003c/h3\u003e\u003cp\u003eA polar point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(r, \\theta)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is converted to Cartesian coordinates by:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex = r\\cos\\theta, \\qquad y = r\\sin\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003esin\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2.9\u003c/mn\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er = 2.9\\cos(5\\theta)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2.9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, we have:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2.9\u003c/mn\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2.9\u003c/mn\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex = 2.9\\cos(5\\theta)\\cos\\theta, \\qquad y = 2.9\\cos(5\\theta)\\sin\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2.9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2.9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003esin\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThis is exactly what is written in the \u003ccode\u003eplot\u003c/code\u003e expression above. Note that TikZ trigonometric functions work in \u003cstrong\u003edegrees\u003c/strong\u003e, so \u003ccode\u003edomain=0:360\u003c/code\u003e directly corresponds to one full revolution.\u003c/p\u003e\n\u003ch2 id=\"experiments-with-tikz-variations\" class=\"headerLink\"\u003e\n \u003ca href=\"#experiments-with-tikz-variations\" class=\"header-mark\" aria-label=\"Direct link to section: Experiments with TikZ Variations\"\u003e\u003c/a\u003eExperiments with TikZ Variations\u003c/h2\u003e\u003cp\u003eThe beauty of the rose curve is that we can change its shape simply by changing the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Here are some variations:\u003c/p\u003e\n\u003ch3 id=\"variation-1-k--3-three-petals\" class=\"headerLink\"\u003e\n \u003ca href=\"#variation-1-k--3-three-petals\" class=\"header-mark\" aria-label=\"Direct link to section: Variation 1: k = 3 (Three Petals)\"\u003e\u003c/a\u003eVariation 1: k = 3 (Three Petals)\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[very thick, blue!75!black, domain=0:360, samples=600, variable=\\t]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e plot (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(3*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*cos(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(3*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*sin(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/rose-curve/rose3.png' srcset=\"/rose-curve/rose3_hu_f10a0b0c99380ebf.webp 800w, /rose-curve/rose3_hu_34867bff7984c610.webp 1200w, /rose-curve/rose3_hu_1983b5feaec39406.webp 1600w\" alt=\"Rose curve with 3 petals (k = 3)\" height=\"415\" width=\"374\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek = 3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (odd) produces \u003cstrong\u003e3 wider petals\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"variation-2-k--7-seven-petals\" class=\"headerLink\"\u003e\n \u003ca href=\"#variation-2-k--7-seven-petals\" class=\"header-mark\" aria-label=\"Direct link to section: Variation 2: k = 7 (Seven Petals)\"\u003e\u003c/a\u003eVariation 2: k = 7 (Seven Petals)\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[very thick, green!60!black, domain=0:360, samples=600, variable=\\t]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e plot (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(7*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*cos(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(7*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*sin(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/rose-curve/rose7.png' srcset=\"/rose-curve/rose7_hu_f61f83d053937291.webp 800w, /rose-curve/rose7_hu_85200257b8c2d4de.webp 1200w, /rose-curve/rose7_hu_fa5e636f2084380e.webp 1600w\" alt=\"Rose curve with 7 petals (k = 7)\" height=\"454\" width=\"444\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek = 7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e produces \u003cstrong\u003e7 slimmer, tighter petals\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"variation-3-k--8-sixteen-petals\" class=\"headerLink\"\u003e\n \u003ca href=\"#variation-3-k--8-sixteen-petals\" class=\"header-mark\" aria-label=\"Direct link to section: Variation 3: k = 8 (Sixteen Petals)\"\u003e\u003c/a\u003eVariation 3: k = 8 (Sixteen Petals)\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[very thick, orange!80!black, domain=0:360, samples=800, variable=\\t]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e plot (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(8*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*cos(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2.9*cos(8*\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)*sin(\u003cspan class=\"k\"\u003e\\t\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/rose-curve/rose8.png' srcset=\"/rose-curve/rose8_hu_b7f9da30eb9c21d1.webp 800w, /rose-curve/rose8_hu_2f862a5d8073322f.webp 1200w, /rose-curve/rose8_hu_e1e7387559ca30a5.webp 1600w\" alt=\"Rose curve with 16 petals (k = 8)\" height=\"464\" width=\"464\"\u003e\u003c/p\u003e\n\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek = 8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cstrong\u003eeven\u003c/strong\u003e, the curve has \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2k = 16\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e petals. The petals are denser, so the number of samples is increased to 800 to keep the curve smooth.\u003c/p\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eDrawing a rose curve with TikZ turns out to be very simple — just a single \u003ccode\u003e\\draw\u003c/code\u003e command with \u003ccode\u003eplot\u003c/code\u003e and a bit of trigonometry. The key is understanding the conversion from polar to Cartesian coordinates:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex = r\\cos\\theta, \\qquad y = r\\sin\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003esin\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eBy varying the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the color, we can produce a wide range of attractive flower shapes, from 3 petals to dozens of petals. This curve is well suited for practicing trigonometric function graphs, polar coordinates, or simply creating mathematical art in LaTeX.\u003c/p\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eWikipedia, \u003cem\u003eRose (mathematics)\u003c/em\u003e — \u003ca href=\"https://en.wikipedia.org/wiki/Rose_%28mathematics%29\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://en.wikipedia.org/wiki/Rose_(mathematics)\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTikZ \u0026amp; PGF Manual — \u003ca href=\"https://tikz.dev\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://tikz.dev\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eRelated article on this blog: \u003ca href=\"/phyllotaxy/\" rel=\"\"\u003eDrawing a Sunflower Pattern (Phyllotaxis) with TikZ\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n", "language": "en" }, { "title" : "TikZ Tutorial for Math Teachers: Drawing Precise Geometry", "date_published" : "2026-08-07T12:00:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/tikz-tutorial/", "url" : "https://www.aantriono.com/en/tikz-tutorial/", "summary": " TikZ Tutorial for Math Teachers: Drawing Precise GeometryTikZ is a LaTeX package for drawing vector graphics directly from code. Unlike pasting images from other applications, TikZ drawings offer:\nHigh precision — coordinates are written as numbers, so dimensions are always exact; Consistency — fonts and drawing styles blend seamlessly with your document; Easy to modify — just edit numbers, no need to redraw; Free — already available on Overleaf with no extra installation. Getting Started on OverleafJust add two lines to your preamble (the part before \\begin{document}):\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch1 id=\"tikz-tutorial-for-math-teachers-drawing-precise-geometry\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-tutorial-for-math-teachers-drawing-precise-geometry\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Tutorial for Math Teachers: Drawing Precise Geometry\"\u003e\u003c/a\u003eTikZ Tutorial for Math Teachers: Drawing Precise Geometry\u003c/h1\u003e\u003cp\u003e\u003cstrong\u003eTikZ\u003c/strong\u003e is a LaTeX package for drawing vector graphics directly from code. Unlike pasting images from other applications, TikZ drawings offer:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eHigh precision\u003c/strong\u003e — coordinates are written as numbers, so dimensions are always exact;\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eConsistency\u003c/strong\u003e — fonts and drawing styles blend seamlessly with your document;\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eEasy to modify\u003c/strong\u003e — just edit numbers, no need to redraw;\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFree\u003c/strong\u003e — already available on Overleaf with no extra installation.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"getting-started-on-overleaf\" class=\"headerLink\"\u003e\n \u003ca href=\"#getting-started-on-overleaf\" class=\"header-mark\" aria-label=\"Direct link to section: Getting Started on Overleaf\"\u003e\u003c/a\u003eGetting Started on Overleaf\u003c/h2\u003e\u003cp\u003eJust add two lines to your preamble (the part before \u003ccode\u003e\\begin{document}\u003c/code\u003e):\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earrows.meta, calc, quotes, backgrounds\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe first line loads the TikZ package. The second loads additional \u003cem\u003elibraries\u003c/em\u003e — extra features like modern arrows, coordinate calculations, and labels. Only load the libraries you need to keep compilation fast.\u003c/p\u003e\n\u003ch2 id=\"tikz-basics\" class=\"headerLink\"\u003e\n \u003ca href=\"#tikz-basics\" class=\"header-mark\" aria-label=\"Direct link to section: TikZ Basics\"\u003e\u003c/a\u003eTikZ Basics\u003c/h2\u003e\u003ch3 id=\"points-and-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#points-and-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Points and Lines\"\u003e\u003c/a\u003ePoints and Lines\u003c/h3\u003e\u003cp\u003eTikZ uses the Cartesian coordinate system: \u003ccode\u003e(x,y)\u003c/code\u003e. Points are drawn with \u003ccode\u003e\\filldraw\u003c/code\u003e, lines with \u003ccode\u003e\\draw\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Points at coordinates (1,2) and (4,3)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e (1,2) circle (3pt);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[red]\u003c/span\u003e (4,3) circle (3pt);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Line from the first point to the second\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick]\u003c/span\u003e (1,2) -- (4,3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/tikz-tutorial/titik_garis.png' srcset=\"/tikz-tutorial/titik_garis_hu_294e19c53b944fd0.webp 800w, /tikz-tutorial/titik_garis_hu_a008488dc2a93ce9.webp 1200w, /tikz-tutorial/titik_garis_hu_5092febd88455ee7.webp 1600w\" alt=\"Basic points and lines\" height=\"137\" width=\"279\"\u003e\u003c/p\u003e\n\u003ch3 id=\"colors-line-thickness-and-styles\" class=\"headerLink\"\u003e\n \u003ca href=\"#colors-line-thickness-and-styles\" class=\"header-mark\" aria-label=\"Direct link to section: Colors, Line Thickness, and Styles\"\u003e\u003c/a\u003eColors, Line Thickness, and Styles\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[red]\u003c/span\u003e (0,0) -- (2,0); \u003cspan class=\"c\"\u003e% thin red line\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue, thick]\u003c/span\u003e (0,1) -- (2,1); \u003cspan class=\"c\"\u003e% thick blue line\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[dashed]\u003c/span\u003e (0,2) -- (2,2); \u003cspan class=\"c\"\u003e% dashed\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[dotted]\u003c/span\u003e (0,3) -- (2,3); \u003cspan class=\"c\"\u003e% dotted\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[-\u0026gt;]\u003c/span\u003e (0,4) -- (2,4); \u003cspan class=\"err\"\u003e%\u003c/span\u003e arrow to the right\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/tikz-tutorial/warna_garis.png' srcset=\"/tikz-tutorial/warna_garis_hu_17682aa2f764e6d2.webp 800w, /tikz-tutorial/warna_garis_hu_5e1b85566cfce5ea.webp 1200w, /tikz-tutorial/warna_garis_hu_4fc9ce7ae9285eef.webp 1600w\" alt=\"Color and line style variations\" height=\"339\" width=\"194\"\u003e\u003c/p\u003e\n\u003ch3 id=\"adding-labels-nodes\" class=\"headerLink\"\u003e\n \u003ca href=\"#adding-labels-nodes\" class=\"header-mark\" aria-label=\"Direct link to section: Adding Labels (Nodes)\"\u003e\u003c/a\u003eAdding Labels (Nodes)\u003c/h3\u003e\u003cp\u003eA node is text attached to a specific point. Label position is set with \u003ccode\u003eabove\u003c/code\u003e, \u003ccode\u003ebelow\u003c/code\u003e, \u003ccode\u003eleft\u003c/code\u003e, \u003ccode\u003eright\u003c/code\u003e, etc.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) 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401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) -- (3,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[above]\u003c/span\u003e at (1.5,0) \u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[below left]\u003c/span\u003e at (0,0) \u003cspan class=\"nb\"\u003e{\u003c/span\u003ebottom left\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[below right]\u003c/span\u003e at (3,0) \u003cspan class=\"nb\"\u003e{\u003c/span\u003ebottom right\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/tikz-tutorial/label_node.png' srcset=\"/tikz-tutorial/label_node_hu_462ef9fb605f1d6.webp 800w, /tikz-tutorial/label_node_hu_fe3767d48d6755cd.webp 1200w, /tikz-tutorial/label_node_hu_cff993140301a6e.webp 1600w\" alt=\"Node labels\" height=\"124\" width=\"555\"\u003e\u003c/p\u003e\n\u003ch2 id=\"case-study-1-the-cartesian-plane\" class=\"headerLink\"\u003e\n \u003ca href=\"#case-study-1-the-cartesian-plane\" class=\"header-mark\" aria-label=\"Direct link to section: Case Study 1: The Cartesian Plane\"\u003e\u003c/a\u003eCase Study 1: The Cartesian Plane\u003c/h2\u003e\u003cp\u003eA classic example: a Cartesian plane with axes, arrows, grid, and number scale.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 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4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[\u0026gt;=Stealth]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% grid\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[gray!40]\u003c/span\u003e (-1,-1) grid (6,6);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% axes\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[-\u0026gt;, thick]\u003c/span\u003e (-1.5,0) -- (6.5,0) node[below left] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ex\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[-\u0026gt;, thick]\u003c/span\u003e (0,-1.5) -- (0,6.5) node[below left] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ey\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% scale\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,5\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e,0.06) -- (\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e,-0.06);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[below]\u003c/span\u003e at (\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e,-0.12) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick]\u003c/span\u003e (0.06,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e) -- (-0.06,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[left]\u003c/span\u003e at (-0.12,\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/tikz-tutorial/kartesius.png' srcset=\"/tikz-tutorial/kartesius_hu_509e8b82b31065d1.webp 800w, /tikz-tutorial/kartesius_hu_ae66d9b5d5ed98a5.webp 1200w, /tikz-tutorial/kartesius_hu_29cf5d298b17b6c5.webp 1600w\" alt=\"Cartesian plane with grid and scale\" height=\"619\" width=\"619\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eTip:\u003c/em\u003e the \u003ccode\u003e\\foreach\u003c/code\u003e command is a \u003cem\u003eloop\u003c/em\u003e — a concise way to repeat a command with different values. It is very useful for scales and grids.\u003c/p\u003e\n\u003ch2 id=\"case-study-2-the-pythagorean-right-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#case-study-2-the-pythagorean-right-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: Case Study 2: The Pythagorean Right Triangle\"\u003e\u003c/a\u003eCase Study 2: The Pythagorean Right Triangle\u003c/h2\u003e\u003cp\u003eCombine all concepts: points, lines, colors, labels, and the right-angle symbol.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[\u0026gt;=Stealth]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% vertices: B at (0,0), A above, C to the right\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (A) at (0,3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (B) at (0,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (C) at (4,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% triangle sides\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[very thick]\u003c/span\u003e (A) -- (B) -- (C) -- cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% right-angle symbol at B\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0.35,0) -- (0.35,0.35) -- (0,0.35);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% vertex labels\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[left]\u003c/span\u003e at (A) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eA\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[below left]\u003c/span\u003e at (B) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eB\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[below right]\u003c/span\u003e at (C) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eC\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% side length labels\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[left]\u003c/span\u003e at (0,1.5) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e6\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[below]\u003c/span\u003e at (2,0) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e8\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[above, sloped]\u003c/span\u003e at (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003eA\u003c/span\u003e\u003cspan class=\"o\"\u003e)!\u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"nb\"\u003e.\u003c/span\u003e\u003cspan class=\"m\"\u003e5\u003c/span\u003e\u003cspan class=\"o\"\u003e!(\u003c/span\u003e\u003cspan class=\"nb\"\u003eC\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e10\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/tikz-tutorial/segitiga.png' srcset=\"/tikz-tutorial/segitiga_hu_43c6cad9ca5d35f7.webp 800w, /tikz-tutorial/segitiga_hu_8d8c4381b1a74313.webp 1200w, /tikz-tutorial/segitiga_hu_8c69771c7755a364.webp 1600w\" alt=\"Pythagorean right triangle\" height=\"314\" width=\"440\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eNote:\u003c/em\u003e \u003ccode\u003e\\coordinate\u003c/code\u003e names a point so it can be reused easily. The expression \u003ccode\u003e$(A)!0.5!(C)$\u003c/code\u003e computes the midpoint between \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e — a feature of the \u003ccode\u003ecalc\u003c/code\u003e library.\u003c/p\u003e\n\u003ch2 id=\"case-study-3-function-graphs\" class=\"headerLink\"\u003e\n \u003ca href=\"#case-study-3-function-graphs\" class=\"header-mark\" aria-label=\"Direct link to section: Case Study 3: Function Graphs\"\u003e\u003c/a\u003eCase Study 3: Function Graphs\u003c/h2\u003e\u003cp\u003eTikZ can also draw function graphs with \u003ccode\u003e\\draw plot\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[\u0026gt;=Stealth, scale=0.8]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% axes\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[-\u0026gt;]\u003c/span\u003e (-0.5,0) -- (4.5,0) node[below] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ex\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[-\u0026gt;]\u003c/span\u003e (0,-0.5) -- (0,9.5) node[left] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ey\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% graph of y = x^2 from x=0 to x=3\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue, very thick, domain=0:3, samples=60]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e plot (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e, \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\x*\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% label\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue, right]\u003c/span\u003e at (3,9) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ey \u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"nb\"\u003e x^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/tikz-tutorial/grafik.png' srcset=\"/tikz-tutorial/grafik_hu_b3725d6d6a41aeff.webp 800w, /tikz-tutorial/grafik_hu_9e47c74d906f3f45.webp 1200w, /tikz-tutorial/grafik_hu_190f07ce497572c1.webp 1600w\" alt=\"Graph of y = x^2\" height=\"634\" width=\"351\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eTip:\u003c/em\u003e \u003ccode\u003edomain=0:3\u003c/code\u003e limits the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e range, \u003ccode\u003esamples=60\u003c/code\u003e sets the number of computed points — the higher the number, the smoother the curve.\u003c/p\u003e\n\u003ch2 id=\"case-study-4-dark-theme\" class=\"headerLink\"\u003e\n \u003ca href=\"#case-study-4-dark-theme\" class=\"header-mark\" aria-label=\"Direct link to section: Case Study 4: Dark Theme\"\u003e\u003c/a\u003eCase Study 4: Dark Theme\u003c/h2\u003e\u003cp\u003eFor presentations or on-screen teaching media, a dark theme can be more comfortable to read.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 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/tikz-tutorial/gelap_hu_2f368c17999bcea4.webp 1600w\" alt=\"Dark theme\" height=\"499\" width=\"499\"\u003e\u003c/p\u003e\n\u003ch2 id=\"quick-reference-of-essential-commands\" class=\"headerLink\"\u003e\n \u003ca href=\"#quick-reference-of-essential-commands\" class=\"header-mark\" aria-label=\"Direct link to section: Quick Reference of Essential Commands\"\u003e\u003c/a\u003eQuick Reference of Essential Commands\u003c/h2\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eCommand\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eFunction\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eExample\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\draw\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003edraw a line\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\draw (0,0) -- (2,2);\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\filldraw\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003efilled point / area\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\filldraw[red] (1,1) circle (2pt);\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\node\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ewrite a label\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\node[above] at (1,1) {A};\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\foreach\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eloop\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\foreach \\n in {1,...,5}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\coordinate\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003ename a point\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\coordinate (A) at (0,0);\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003eplot\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003efunction graph\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003eplot (\\x, {\\x*\\x});\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eTikZ may look intimidating at first, but once you master five basic concepts — points, lines, nodes, colors, and \u003ccode\u003eforeach\u003c/code\u003e — you can draw almost any geometry for worksheets, exam questions, or presentation slides.\u003c/p\u003e\n\u003cp\u003eThe key to practice: \u003cstrong\u003estart small\u003c/strong\u003e, compile often, and look at the results immediately. Overleaf shows output in real time, so the learning process feels fast.\u003c/p\u003e\n\u003ch2 id=\"references\" class=\"headerLink\"\u003e\n \u003ca href=\"#references\" class=\"header-mark\" aria-label=\"Direct link to section: References\"\u003e\u003c/a\u003eReferences\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003eOfficial TikZ documentation: \u003ca href=\"https://ctan.org/pkg/pgf\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ectan.org/pkg/pgf\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eExample gallery: \u003ca href=\"https://texample.net\" target=\"_blank\" rel=\"noopener noreferrer\"\u003etexample.net\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eLaTeX wiki: \u003ca href=\"https://overleaf.com/learn\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eoverleaf.com/learn\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eRelated article on this blog: \u003ca href=\"/rose-curve/\" rel=\"\"\u003eDrawing a Rose Curve with TikZ\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n", "language": "en" }, { "title" : "In Times of Fitnah, Hold Fast to the Rope of Allah", "date_published" : "2026-07-28T03:44:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/zaman-fitnah-berpegang-pada-tali-allah/", "url" : "https://www.aantriono.com/en/zaman-fitnah-berpegang-pada-tali-allah/", "summary": " IntroductionLiving in the end times presents immensely complex spiritual and social challenges. Waves of tribulation (fitnah) strike continuously from all directions—manifesting as sectarian disunity, ideological deviations, misleading narratives, fabricated lies, and divisive accusations among Muslims. In such turbulent waters, individuals often lose their moral compass and drift into harmful conflicts.\nWhere should a believer turn when encircled by fitnah? The answer lies in the eternal legacy of the Prophet Muhammad (ﷺ): the Qur\u0026rsquo;an and the Sunnah. Holding fast to the Rope of Allah (Hablullah) is the sole path of salvation (najat) capable of guiding a servant through the storms of end-time trials.\nBelow is an in-depth reflection on the necessity of returning to the Rope of Allah during times of turmoil.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eLiving in the end times presents immensely complex spiritual and social challenges. Waves of tribulation (\u003cem\u003efitnah\u003c/em\u003e) strike continuously from all directions—manifesting as sectarian disunity, ideological deviations, misleading narratives, fabricated lies, and divisive accusations among Muslims. In such turbulent waters, individuals often lose their moral compass and drift into harmful conflicts.\u003c/p\u003e\n\u003cp\u003eWhere should a believer turn when encircled by \u003cem\u003efitnah\u003c/em\u003e? The answer lies in the eternal legacy of the Prophet Muhammad (ﷺ): \u003cstrong\u003ethe Qur\u0026rsquo;an and the Sunnah\u003c/strong\u003e. Holding fast to the Rope of Allah (\u003cem\u003eHablullah\u003c/em\u003e) is the sole path of salvation (\u003cem\u003enajat\u003c/em\u003e) capable of guiding a servant through the storms of end-time trials.\u003c/p\u003e\n\u003cp\u003eBelow is an in-depth reflection on the necessity of returning to the Rope of Allah during times of turmoil.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"1-the-quran-and-sunnah-the-anchor-of-salvation\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-the-quran-and-sunnah-the-anchor-of-salvation\" class=\"header-mark\" aria-label=\"Direct link to section: 1. The Qur\u0026rsquo;an and Sunnah: The Anchor of Salvation\"\u003e\u003c/a\u003e1. The Qur\u0026rsquo;an and Sunnah: The Anchor of Salvation\u003c/h2\u003e\u003cp\u003eIslam teaches that whenever conflict, deviation, or confusion arises within the community, the solution is not to surrender to desires, hatred, or factional pride, but to refer all matters back to Allah and His Messenger.\u003c/p\u003e\n\u003cp\u003eAllah Almighty says:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eيَا أَيُّهَا الَّذِينَ آمَنُوا أَطِيعُوا اللَّهَ وَأَطِيعُوا الرَّسُولَ وَأُولِي الْأَمْرِ مِنكُمْ ۖ فَإِن تَنَازَعْتُمْ فِي شَيْءٍ فَرُدُّوهُ إِلَى اللَّهِ وَالرَّسُولِ إِن كُنتُمْ تؤْمِنُونَ بِاللَّهِ وَالْيَوْمِ الْآخِرِ ۚ ذَٰلِك خَيْرٌ وَأَحْسَنُ تَأْوِيلًا\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTranslation:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;O you who have believed, obey Allah and obey the Messenger and those in authority among you. And if you disagree over anything, refer it to Allah and the Messenger, if you should believe in Allah and the Last Day. That is the best [way] and best in result.\u0026rdquo;\u003c/em\u003e (Surah An-Nisa [4]: 59)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe Prophet Muhammad (ﷺ) guaranteed protection from straying when adhering to these two sources:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;I have left among you two matters which if you adhere to them, you will never go astray: the Book of Allah and the Sunnah of His Prophet.\u0026rdquo;\u003c/em\u003e (Narrated by Al-Hakim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"2-unity-under-the-banner-of-hablullah\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-unity-under-the-banner-of-hablullah\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Unity Under the Banner of Hablullah\"\u003e\u003c/a\u003e2. Unity Under the Banner of \u003cem\u003eHablullah\u003c/em\u003e\u003c/h2\u003e\u003cp\u003eDivision and enmity only weaken the fabric of the Muslim community. Allah SWT commands believers to unite under the divine bond of religion (\u003cem\u003eHablullah\u003c/em\u003e) and forbids splintering into factions:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَاعْتَصِمُوا بِحَبْلِ اللَّهِ جَمِيعًا وَلَا تَفَرَّقُوا ۚ وَاذْكُرُوا نِعْمَتَ اللَّهِ عَلَيْكُمْ إِذْ كُنتُمْ أَعْدَاءً فَأَلَّفَ بَيْنَ قُلُوبِكُمْ فَأَصْبَحْتُم بِنِعْمَتِهِ إِخْوَانًا\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTranslation:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And hold firmly to the rope of Allah all together and do not become divided. And remember the favor of Allah upon you - when you were enemies and He brought your hearts together and you became, by His favor, brothers\u0026hellip;\u0026rdquo;\u003c/em\u003e (Surah Ali \u0026lsquo;Imran [3]: 103)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eFaith is the ultimate bond that unites human hearts across diverse backgrounds while rescuing them from the brink of destruction.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"3-navigating-fitnah-with-wisdom-and-patience\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-navigating-fitnah-with-wisdom-and-patience\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Navigating Fitnah with Wisdom and Patience\"\u003e\u003c/a\u003e3. Navigating \u003cem\u003eFitnah\u003c/em\u003e with Wisdom and Patience\u003c/h2\u003e\u003cp\u003eFaced with worldly deception and injustice, believers are instructed to maintain wisdom, composure, and emotional restraint:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eAvoid Unrest and Harm:\u003c/strong\u003e Refrain from participating in chaos, violence, or actions that destroy societal peace.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eExercise Patience:\u003c/strong\u003e Trust that Allah is never unaware nor deceivable. Whoever engages in deceit or oppression will ultimately face divine justice in the Hereafter.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFocus on Self-Correction:\u003c/strong\u003e Concentrate on personal reform (\u003cem\u003eishlah\u003c/em\u003e) and constructive efforts within one\u0026rsquo;s capability, rather than exhausting spiritual energy on negativity and hatred.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"4-recognizing-salvation-as-divine-grace\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-recognizing-salvation-as-divine-grace\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Recognizing Salvation as Divine Grace\"\u003e\u003c/a\u003e4. Recognizing Salvation as Divine Grace\u003c/h2\u003e\u003cp\u003eNo human being is immune to errors or entitled to self-righteousness. Whenever we are protected from wrongdoing, straying, or division, it is not due to our own intellect or superiority, but purely through \u003cstrong\u003eAllah\u0026rsquo;s guidance, grace, and divine protection\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eTherefore, alongside striving to rectify our actions, we must continuously pray for the well-being of the community and seek a noble final ending (\u003cem\u003ehusnul khatimah\u003c/em\u003e).\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eOvercoming the trials of the end times demands steadfast faith and a clean heart:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eReturn to Sacred Guidance:\u003c/strong\u003e Let the Qur\u0026rsquo;an and Sunnah be the ultimate judge in disputes.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eStrengthen Brotherhood:\u003c/strong\u003e Prioritize unity upon the Rope of Allah above factional interests.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMaintain Inner Peace:\u003c/strong\u003e Face worldly trials with patience, sincerity, and trust in God.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eMay Allah SWT grant us the strength to remain firmly attached to His Rope and bestow upon us peace and salvation in this life and the next. \u003cem\u003eÂmîn.\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the original article by Ibnu Anwar (via Eramuslim).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Advice of Sheikh Abdul Qadir Al-Jilani to Those Facing Hardship", "date_published" : "2026-07-28T03:38:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/nasihat-syekh-abdul-qadir-al-jilani-hidup-susah/", "url" : "https://www.aantriono.com/en/nasihat-syekh-abdul-qadir-al-jilani-hidup-susah/", "summary": " IntroductionLife in this world is an ever-revolving cycle of trials. While there are moments of ease and abundance, a servant is frequently tested with hardship, poverty, and overwhelming distress. During such dark periods, the human spirit can easily waver, sink into sorrow, or fall into negative assumptions about the Almighty.\nIn his legendary spiritual masterpiece, Futuh Al-Ghaib (The Revelations of the Unseen), the great scholar and master of spiritual wayfarers, Sheikh Abdul Qadir Al-Jilani (rahimahullah), offers profound, tender, and deeply monotheistic advice for those navigating the stormy seas of worldly adversity.\nBelow is an insightful reflection on Sheikh Abdul Qadir Al-Jilani\u0026rsquo;s golden counsel for anyone seeking inner peace amidst life\u0026rsquo;s struggles.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eLife in this world is an ever-revolving cycle of trials. While there are moments of ease and abundance, a servant is frequently tested with hardship, poverty, and overwhelming distress. During such dark periods, the human spirit can easily waver, sink into sorrow, or fall into negative assumptions about the Almighty.\u003c/p\u003e\n\u003cp\u003eIn his legendary spiritual masterpiece, \u003cem\u003eFutuh Al-Ghaib\u003c/em\u003e (The Revelations of the Unseen), the great scholar and master of spiritual wayfarers, \u003cstrong\u003eSheikh Abdul Qadir Al-Jilani\u003c/strong\u003e (\u003cem\u003erahimahullah\u003c/em\u003e), offers profound, tender, and deeply monotheistic advice for those navigating the stormy seas of worldly adversity.\u003c/p\u003e\n\u003cp\u003eBelow is an insightful reflection on Sheikh Abdul Qadir Al-Jilani\u0026rsquo;s golden counsel for anyone seeking inner peace amidst life\u0026rsquo;s struggles.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"1-refraining-from-blaming-god-during-hardship\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-refraining-from-blaming-god-during-hardship\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Refraining from Blaming God During Hardship\"\u003e\u003c/a\u003e1. Refraining from Blaming God During Hardship\u003c/h2\u003e\u003cp\u003eSheikh Abdul Qadir Al-Jilani opens his counsel with a firm yet compassionate warning to the poor and distressed: never complain or blame Allah for your earthly condition.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;Do not say, O unfortunate one! Never say that Allah has made you poor, turned the world away from you, humiliated you, or neglected to suffice you in this life\u0026hellip;\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eIt is human nature to compare one\u0026rsquo;s status with others. One might see how God bestows wealth, status, and ease upon others day and night, even though both share the same faith, worship the same Lord, and descend from the same common ancestors, Adam and Eve. Yet, Sheikh Abdul Qadir cautions believers against entertaining ill thoughts (\u003cem\u003esu\u0026rsquo;uzhan\u003c/em\u003e) toward their Creator.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"2-hidden-divine-wisdom-nurturing-faith-and-patience\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-hidden-divine-wisdom-nurturing-faith-and-patience\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Hidden Divine Wisdom: Nurturing Faith and Patience\"\u003e\u003c/a\u003e2. Hidden Divine Wisdom: Nurturing Faith and Patience\u003c/h2\u003e\u003cp\u003eWhy does Allah test a faithful believer with hardship and poverty? Sheikh Abdul Qadir Al-Jilani explains the inner spiritual wisdom in poetic prose:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;Indeed, Allah has treated you thus because your innate nature (fitrah) is pure, and the cooling breeze of divine mercy continuously showers upon you in the form of patience, contentment (ridha), and spiritual gnosis (ma\u0026rsquo;rifah). The light of faith and monotheism shines upon you, strengthening the roots, seeds, branches, and fruits of your tree of faith until its shade spreads far and wide.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eHardship for a believer is not a sign of divine anger, but a form of \u003cstrong\u003eDivine refinement\u003c/strong\u003e. Allah purifies the soul from pride, arrogance, and worldly attachment (\u003cem\u003ehubbud dunya\u003c/em\u003e). Through the fire of adversity, Allah shapes the believer\u0026rsquo;s spirit to become resilient, deeply rooted in patience, and steadfast in faith.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"3-the-certainty-of-sustenance-and-decree\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-the-certainty-of-sustenance-and-decree\" class=\"header-mark\" aria-label=\"Direct link to section: 3. The Certainty of Sustenance and Decree\"\u003e\u003c/a\u003e3. The Certainty of Sustenance and Decree\u003c/h2\u003e\u003cp\u003eA major source of human anxiety is excessive worry over daily sustenance (\u003cem\u003erizq\u003c/em\u003e) and envy toward others\u0026rsquo; success. Regarding this, Sheikh Abdul Qadir establishes a timeless spiritual rule:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eWhat is decreed for you:\u003c/strong\u003e It will inevitably reach you, and you will be led toward it, so that your meeting with your allocated sustenance is guaranteed at the exact appointed time.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eWhat is not decreed for you:\u003c/strong\u003e You will be kept away from it, and it will be kept away from you, so that you and it shall never meet.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eTherefore, he advises:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;Do not be greedy for what is destined for you, nor be anxious about it. And do not feel regret over what was intended for someone else.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"4-quranic-guidance-on-guarding-ones-eyes\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-quranic-guidance-on-guarding-ones-eyes\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Qur\u0026rsquo;anic Guidance on Guarding One\u0026rsquo;s Eyes\"\u003e\u003c/a\u003e4. Qur\u0026rsquo;anic Guidance on Guarding One\u0026rsquo;s Eyes\u003c/h2\u003e\u003cp\u003eRegarding the prohibition of envying the material grandeur of others and the virtue of contentment (\u003cem\u003eqana\u0026rsquo;ah\u003c/em\u003e), Allah SWT states in the Holy Qur\u0026rsquo;an:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَلَا تَمُدَّنَّ عَيْنَيْكَ إِلَىٰ مَا مَتَّعْنَا بِهِ أَزْوَاجًا مِّنْهُمْ زَهْرَةَ الْحَيَاةِ الدُّنْيَا لِنَفْتِنَهُمْ فِيهِ ۚ وَرِزْقُ رَبِّكَ خَيْرٌ وَأَبْقَىٰ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTranslation:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And do not extend your eyes toward that by which We have given enjoyment to [certain] categories of them, [it being but] the splendor of worldly life by which We test them. And the provision of your Lord is better and more enduring.\u0026rdquo;\u003c/em\u003e (QS. Taha [20]: 131)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eWorldly luxury given to others is often a trial (\u003cem\u003efitnah\u003c/em\u003e). In contrast, peace of mind, contentment with divine destiny, and the rewards of the Hereafter are far more noble and lasting.\u003c/p\u003e\n\u003cp\u003eFor those who remain patient and content, Allah promises an unimaginable reward:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eفَلَا تَعْلَمُ نَفْسٌ مَّا أُخْفِيَ لَهُم مِّن قُرَّةِ أَعْيُنٍ جَزَاءً بِمَا كَانُوا يَعْمَلُونَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTranslation:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And no soul knows what has been hidden for them of comfort for eyes as reward for what they used to do.\u0026rdquo;\u003c/em\u003e (QS. As-Sajdah [32]: 17)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"conclusion-and-spiritual-takeaway\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion-and-spiritual-takeaway\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion and Spiritual Takeaway\"\u003e\u003c/a\u003eConclusion and Spiritual Takeaway\u003c/h2\u003e\u003cp\u003eSheikh Abdul Qadir Al-Jilani\u0026rsquo;s counsel in \u003cem\u003eFutuh Al-Ghaib\u003c/em\u003e provides a serene spiritual sanctuary for anyone facing adversity:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003ePurify Your Thoughts:\u003c/strong\u003e Realize that hardship is not humiliation, but a means of spiritual elevation.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eEmbrace Contentment (\u003cem\u003eRidha\u003c/em\u003e):\u003c/strong\u003e Surrendering with peace of heart to God\u0026rsquo;s decree attracts divine grace.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFocus on Devotion:\u003c/strong\u003e No act is more beloved to Allah than fulfilling worship, avoiding sin, and remaining content with your divine portion.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eMay Allah grant us tranquility of heart, strength of faith, and divine contentment in every decree. \u003cem\u003eÂmîn.\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from Futuh Al-Ghaib by Sheikh Abdul Qadir Al-Jilani (via Eramuslim).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Pearls of Wisdom by Ibn 'Atha'illah: Relying on Deeds vs. Divine Mercy", "date_published" : "2026-07-28T03:12:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mutiara-hikmah-ibn-athaillah/", "url" : "https://www.aantriono.com/en/mutiara-hikmah-ibn-athaillah/", "summary": " IntroductionThe book Al-Hikam al-‘Athâ’iyyah by Sheikh Ibn \u0026lsquo;Atha\u0026rsquo;illah as-Sakandari is a monumental masterpiece in Islamic Sufism. Renowned for its exquisite literary elegance, concise rhetoric, and profound spiritual depth, Ibn \u0026lsquo;Atha\u0026rsquo;illah masterfully weaves imagery and linguistic finesse to captivate readers and spiritual seekers across generations.\nTo fully grasp the wisdom enshrined within this text, a multidimensional approach is essential—combining linguistic analysis with the classic commentaries (syarah) of prominent scholars such as Ibn \u0026lsquo;Abbad al-Randi, Sheikh Abdullah Syarqawi, Ibn \u0026lsquo;Ajibah, Ahmad Zarruq, and Dr. Muhammad Sa\u0026rsquo;id Ramadhan al-Buthi.\nBelow is an analytical commentary on the First Pearl of Wisdom from the book Al-Hikam.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThe book \u003cem\u003eAl-Hikam al-‘Athâ’iyyah\u003c/em\u003e by Sheikh Ibn \u0026lsquo;Atha\u0026rsquo;illah as-Sakandari is a monumental masterpiece in Islamic Sufism. Renowned for its exquisite literary elegance, concise rhetoric, and profound spiritual depth, Ibn \u0026lsquo;Atha\u0026rsquo;illah masterfully weaves imagery and linguistic finesse to captivate readers and spiritual seekers across generations.\u003c/p\u003e\n\u003cp\u003eTo fully grasp the wisdom enshrined within this text, a multidimensional approach is essential—combining linguistic analysis with the classic commentaries (\u003cem\u003esyarah\u003c/em\u003e) of prominent scholars such as Ibn \u0026lsquo;Abbad al-Randi, Sheikh Abdullah Syarqawi, Ibn \u0026lsquo;Ajibah, Ahmad Zarruq, and Dr. Muhammad Sa\u0026rsquo;id Ramadhan al-Buthi.\u003c/p\u003e\n\u003cp\u003eBelow is an analytical commentary on the \u003cstrong\u003eFirst Pearl of Wisdom\u003c/strong\u003e from the book \u003cem\u003eAl-Hikam\u003c/em\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-text-and-translation-of-the-1st-wisdom\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-text-and-translation-of-the-1st-wisdom\" class=\"header-mark\" aria-label=\"Direct link to section: The Text and Translation of the 1st Wisdom\"\u003e\u003c/a\u003eThe Text and Translation of the 1st Wisdom\u003c/h2\u003e\u003cp\u003eSheikh Ibn \u0026lsquo;Atha\u0026rsquo;illah as-Sakandari opens his masterpiece with this fundamental aphorism:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eمِنْ عَلَامَةِ الِاعْتِمَادِ عَلَى الْعَمَلِ – نُقْصَانُ الرَّجَاءِ عِنْدَ وُجُودِ الزَّلَلِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTranslation:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;One of the signs of relying on one\u0026rsquo;s own deeds is the diminution of hope (in God\u0026rsquo;s mercy) when a slip or sin occurs.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"linguistic-and-rhetorical-analysis-balaghah\" class=\"headerLink\"\u003e\n \u003ca href=\"#linguistic-and-rhetorical-analysis-balaghah\" class=\"header-mark\" aria-label=\"Direct link to section: Linguistic and Rhetorical Analysis (Balaghah)\"\u003e\u003c/a\u003eLinguistic and Rhetorical Analysis (\u003cem\u003eBalaghah\u003c/em\u003e)\u003c/h2\u003e\u003cp\u003eFrom a stylistic and linguistic standpoint, this aphorism is meticulously crafted:\u003c/p\u003e\n\u003ch3 id=\"1-the-rhetorical-style-of-al-taqdîm-wa-al-takhîr\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-the-rhetorical-style-of-al-taqd%c3%aem-wa-al-takh%c3%aer\" class=\"header-mark\" aria-label=\"Direct link to section: 1. The Rhetorical Style of Al-Taqdîm wa Al-Ta’khîr\"\u003e\u003c/a\u003e1. The Rhetorical Style of \u003cem\u003eAl-Taqdîm wa Al-Ta’khîr\u003c/em\u003e\u003c/h3\u003e\u003cp\u003eThe sentence utilizes \u003cem\u003eal-taqdîm wa al-ta’khîr\u003c/em\u003e (fronting and postponing), placing the predicate (\u003cem\u003ekhabar\u003c/em\u003e) \u003ccode\u003eمِنْ عَلَامَةِ\u003c/code\u003e ahead of the subject (\u003cem\u003emubtada’\u003c/em\u003e) \u003ccode\u003eنُقْصَانُ الرَّجَاءِ\u003c/code\u003e. This syntactic arrangement immediately draws the reader\u0026rsquo;s attention to the core diagnostic indicator of spiritual reliance.\u003c/p\u003e\n\u003ch3 id=\"2-the-partitive-preposition-min-li-al-tabîdh\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-the-partitive-preposition-min-li-al-tab%c3%aedh\" class=\"header-mark\" aria-label=\"Direct link to section: 2. The Partitive Preposition Min (Li al-Tab’îdh)\"\u003e\u003c/a\u003e2. The Partitive Preposition \u003cem\u003eMin\u003c/em\u003e (\u003cem\u003eLi al-Tab’îdh\u003c/em\u003e)\u003c/h3\u003e\u003cp\u003eThe preposition \u003cstrong\u003e\u0026ldquo;min\u0026rdquo;\u003c/strong\u003e serves a partitive function (\u003cem\u003eli al-tab\u0026rsquo;idh\u003c/em\u003e). It signifies that losing hope in Allah\u0026rsquo;s mercy upon committing a sin is only \u003cstrong\u003eone\u003c/strong\u003e indicator among several signs of excessive reliance on self-effort.\u003c/p\u003e\n\u003ch3 id=\"3-semiotics-and-the-concept-of-\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-semiotics-and-the-concept-of-\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Semiotics and the Concept of \u0026lsquo;Alâmah\"\u003e\u003c/a\u003e3. Semiotics and the Concept of \u003cem\u003e\u0026lsquo;Alâmah\u003c/em\u003e\u003c/h3\u003e\u003cp\u003eThe term \u003cem\u003e‘alâmah\u003c/em\u003e denotes a distinct sign or symbol signaling the existence of a condition. In modern linguistics, this relates directly to semiotics (\u003cem\u003e‘ilm al-‘alâmah\u003c/em\u003e), examining outward signs to uncover inner realities.\u003c/p\u003e\n\u003ch3 id=\"4-etymology-of-al-itimâd-and-classical-arabic-proverb\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-etymology-of-al-itim%c3%a2d-and-classical-arabic-proverb\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Etymology of Al-I’timâd and Classical Arabic Proverb\"\u003e\u003c/a\u003e4. Etymology of \u003cem\u003eAl-I’timâd\u003c/em\u003e and Classical Arabic Proverb\u003c/h3\u003e\u003cp\u003eDerived from \u003cem\u003ei’tamada – ya’tamidu\u003c/em\u003e (synonymous with \u003cem\u003eal-ittikâl\u003c/em\u003e), \u003cem\u003eal-i’timâd\u003c/em\u003e means to depend or lean upon. Relying foolishly on something is illustrated by a classic Arabic proverb:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eلَا تَشْرَبْ السَمَّ اِتِّكَالاً عَلَى التِرْيَاق\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026ldquo;Do not drink poison relying solely on the antidote.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThis proverb warns against presumption—committing wrongdoings carelessly while overly relying on expected forgiveness.\u003c/p\u003e\n\u003ch3 id=\"5-definitions-of-al-amal-ar-rajâ-and-al-zalal\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-definitions-of-al-amal-ar-raj%c3%a2-and-al-zalal\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Definitions of Al-‘Amal, Ar-Rajâ’, and Al-Zalal\"\u003e\u003c/a\u003e5. Definitions of \u003cem\u003eAl-‘Amal\u003c/em\u003e, \u003cem\u003eAr-Rajâ’\u003c/em\u003e, and \u003cem\u003eAl-Zalal\u003c/em\u003e\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eAl-‘Amal:\u003c/strong\u003e All acts of devotion (prayer, charity, remembrance) performed to seek closeness to Allah.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAr-Rajâ’:\u003c/strong\u003e Etymologically means hope; terminologically refers to the heart\u0026rsquo;s attachment to expected divine grace in the future.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAl-Zalal:\u003c/strong\u003e Refers to a slip, lapse, or sin (\u003cem\u003eal-itsm / ad-dzanb\u003c/em\u003e).\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"6-aesthetic-assonance-as-saju\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-aesthetic-assonance-as-saju\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Aesthetic Assonance (As-Saj’u)\"\u003e\u003c/a\u003e6. Aesthetic Assonance (\u003cem\u003eAs-Saj’u\u003c/em\u003e)\u003c/h3\u003e\u003cp\u003eThe aphorism employs \u003cem\u003eas-saj’u\u003c/em\u003e (rhyming assonance) between \u003ccode\u003eالْعَمَل\u003c/code\u003e (\u003cem\u003eal-‘amal\u003c/em\u003e) and \u003ccode\u003eالزَّلَل\u003c/code\u003e (\u003cem\u003eal-zalal\u003c/em\u003e). This phonetic harmony creates a pleasing musicality that leaves a lasting impression on the listener.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"spiritual-depth-and-commentary\" class=\"headerLink\"\u003e\n \u003ca href=\"#spiritual-depth-and-commentary\" class=\"header-mark\" aria-label=\"Direct link to section: Spiritual Depth and Commentary\"\u003e\u003c/a\u003eSpiritual Depth and Commentary\u003c/h2\u003e\u003cp\u003eThrough this concise aphorism, Sheikh Ibn \u0026lsquo;Atha\u0026rsquo;illah guides spiritual seekers (\u003cem\u003esalik\u003c/em\u003e) toward a refined state of heart:\u003c/p\u003e\n\u003ch3 id=\"1-correcting-the-hearts-reliance-i\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-correcting-the-hearts-reliance-i\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Correcting the Heart\u0026rsquo;s Reliance (I\u0026rsquo;timad)\"\u003e\u003c/a\u003e1. Correcting the Heart\u0026rsquo;s Reliance (\u003cem\u003eI\u0026rsquo;timad\u003c/em\u003e)\u003c/h3\u003e\u003cp\u003eThis wisdom is not an encouragement to abandon worship or disregard good deeds. Ibn \u0026lsquo;Atha\u0026rsquo;illah himself was a devout practitioner throughout his life. Rather, the lesson forbids spiritual pride—relying on one\u0026rsquo;s deeds as earned merit rather than placing ultimate trust in Allah\u0026rsquo;s boundless grace (\u003cem\u003efadhl\u003c/em\u003e) and mercy (\u003cem\u003erahmah\u003c/em\u003e).\u003c/p\u003e\n\u003cp\u003eWhen a person relies solely on deeds, committing a sin causes their hope (\u003cem\u003eraja\u0026rsquo;\u003c/em\u003e) to collapse into despair. Conversely, one whose hope rests upon Allah\u0026rsquo;s mercy remains resilient, seeking repentance immediately without losing faith.\u003c/p\u003e\n\u003ch3 id=\"2-divine-mercy-as-the-source-of-salvation\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-divine-mercy-as-the-source-of-salvation\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Divine Mercy as the Source of Salvation\"\u003e\u003c/a\u003e2. Divine Mercy as the Source of Salvation\u003c/h3\u003e\u003cp\u003eThis aligns with the noble Hadith of the Prophet Muhammad (ﷺ):\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;None of your deeds will enter you into Paradise.\u0026rdquo;\u003c/em\u003e The companions asked: \u003cem\u003e\u0026ldquo;Not even you, O Messenger of Allah?\u0026rdquo;\u003c/em\u003e He replied: \u003cem\u003e\u0026ldquo;Not even me, unless Allah encompasses me with His grace and mercy.\u0026rdquo;\u003c/em\u003e (Sahih al-Bukhari 5673, Sahih Muslim 2816).\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"3-rectifying-the-transactional-mindset\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-rectifying-the-transactional-mindset\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Rectifying the Transactional Mindset\"\u003e\u003c/a\u003e3. Rectifying the Transactional Mindset\u003c/h3\u003e\u003cp\u003eAs noted by Dr. Muhammad Sa\u0026rsquo;id Ramadhan al-Buthi, it is improper to treat Paradise as a commercial purchase bought through prayers and fasting. The ability to stand in prayer, fast, or believe is itself a divine grant (\u003cem\u003etaufiq\u003c/em\u003e) from Allah.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe first wisdom of Ibn \u0026lsquo;Atha\u0026rsquo;illah teaches us to purify our intentions. While we must strive earnestly in worship as a duty of servitude, our ultimate trust, hope, and peace of mind must rest entirely upon the unearned grace and mercy of Almighty Allah.\u003c/p\u003e\n\u003cp\u003eMay Allah guide us to be among those who perform righteous deeds with sincerity and constantly seek His divine mercy. \u003cem\u003eÂmîn.\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the original article by Prof. Dr. H. Syihabuddin Qalyubi, Lc. M.Ag. (Professor at Faculty of Adab and Cultural Sciences, UIN Sunan Kalijaga Yogyakarta).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Unila Community Service and Central Lampung Middle School Math MGMP: MathCityMap-Based Deep Learning Training", "date_published" : "2026-07-23T08:00:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pelatihan-deep-learning-mathcitymap/", "url" : "https://www.aantriono.com/en/pelatihan-deep-learning-mathcitymap/", "summary": "In an ongoing effort to transform mathematics learning into a more meaningful, interactive, and relevant experience for daily life, the Community Service Team (PkM) from Universitas Lampung (Unila) partnered with the Middle School Mathematics Teacher Subject Forum (MGMP) of Central Lampung Regency to hold an intensive two-day training workshop.\nThe workshop, titled \u0026ldquo;Development of Technology-Based MathCityMap Deep Learning Math Trail Activities to Enhance Middle School Teacher Competence in 2026\u0026rdquo;, took place on Tuesday–Wednesday, July 21–22, 2026, at SMPN 4 Terbanggi Besar. The event was designed to bridge abstract mathematical theory with the physical reality of the school environment through location-based technology.\nSpeech and Official Opening of the Training Photo 1: Group picture with Mrs. Devira Santi, S.T., M.T. (Head of Basic Education Division, Disdikbud Central Lampung) while providing guidance and officially opening the training.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eIn an ongoing effort to transform mathematics learning into a more meaningful, interactive, and relevant experience for daily life, the Community Service Team (PkM) from Universitas Lampung (Unila) partnered with the Middle School Mathematics Teacher Subject Forum (MGMP) of Central Lampung Regency to hold an intensive two-day training workshop.\u003c/p\u003e\n\u003cp\u003eThe workshop, titled \u003cstrong\u003e\u0026ldquo;Development of Technology-Based MathCityMap Deep Learning Math Trail Activities to Enhance Middle School Teacher Competence in 2026\u0026rdquo;\u003c/strong\u003e, took place on Tuesday–Wednesday, July 21–22, 2026, at SMPN 4 Terbanggi Besar. The event was designed to bridge abstract mathematical theory with the physical reality of the school environment through location-based technology.\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/pelatihan-deep-learning-mathcitymap/foto-1.webp\" title=\"Training Opening by Head of Basic Education Disdikbud Central Lampung\" data-thumbnail=\"/pelatihan-deep-learning-mathcitymap/foto-1.webp\" data-sub-html=\"\u003ch2\u003eSpeech and Official Opening of the Training\u003c/h2\u003e\u003cp\u003eTraining Opening by Head of Basic Education Disdikbud Central Lampung\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/pelatihan-deep-learning-mathcitymap/foto-1.webp' srcset=\"/pelatihan-deep-learning-mathcitymap/foto-1_hu_60501bc05e3c529c.webp 800w, /pelatihan-deep-learning-mathcitymap/foto-1_hu_d166e27d3d4ce4bf.webp 1200w, /pelatihan-deep-learning-mathcitymap/foto-1_hu_3eeb0a9c94a739dd.webp 1600w\" alt=\"Training Opening by Head of Basic Education Disdikbud Central Lampung\" height=\"720\" width=\"1280\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSpeech and Official Opening of the Training\u003c/figcaption\u003e\n\u003c/figure\u003e\n\u003cem\u003ePhoto 1: Group picture with Mrs. Devira Santi, S.T., M.T. (Head of Basic Education Division, Disdikbud Central Lampung) while providing guidance and officially opening the training.\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"official-opening-and-strategic-foundation\" class=\"headerLink\"\u003e\n \u003ca href=\"#official-opening-and-strategic-foundation\" class=\"header-mark\" aria-label=\"Direct link to section: Official Opening and Strategic Foundation\"\u003e\u003c/a\u003eOfficial Opening and Strategic Foundation\u003c/h2\u003e\u003cp\u003eThe event was officially opened by \u003cstrong\u003eMrs. Devira Santi, S.T., M.T.\u003c/strong\u003e, Head of Basic Education at the Department of Education and Culture of Central Lampung Regency. In her opening remarks, she emphasized the critical importance of technology adaptation at the junior high school level.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;Mathematics education can no longer remain trapped in rigid paper formulas. A deep learning approach combined with open-source technology like MathCityMap is a strategic step toward strengthening students\u0026rsquo; reasoning and numeracy literacy in Central Lampung Regency,\u0026rdquo;\u003c/em\u003e she stated.\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe training was attended by \u003cstrong\u003e50 middle school mathematics teachers\u003c/strong\u003e from various active member schools across the Central Lampung Math MGMP network.\u003c/p\u003e\n\u003cp\u003eThe Unila PkM Team facilitating the entire series of activities was led by \u003cstrong\u003eDr. Chika Rahayu, M.Pd.\u003c/strong\u003e, accompanied by experienced lecturers and facilitators:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eHesti Noviyana, M.Pd.\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFadil Firdian, M.Pd.T.\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eElora Theona br. Sibuea\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eJeova Jire\u003c/strong\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"urgency-of-deep-learning-and-the-mathcitymap-ecosystem\" class=\"headerLink\"\u003e\n \u003ca href=\"#urgency-of-deep-learning-and-the-mathcitymap-ecosystem\" class=\"header-mark\" aria-label=\"Direct link to section: Urgency of Deep Learning and the MathCityMap Ecosystem\"\u003e\u003c/a\u003eUrgency of \u003cem\u003eDeep Learning\u003c/em\u003e and the \u003cem\u003eMathCityMap\u003c/em\u003e Ecosystem\u003c/h2\u003e\u003cp\u003eWhy are \u003cem\u003eDeep Learning\u003c/em\u003e and \u003cem\u003eMathCityMap\u003c/em\u003e the primary focus?\u003c/p\u003e\n\u003cp\u003eDeep Learning in mathematics requires students not merely to memorize problem-solving algorithms, but to intuitively understand concepts, connect ideas, and apply them to real-world situations.\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/pelatihan-deep-learning-mathcitymap/foto-2.webp\" title=\"Deep Learning Material Presentation Session\" data-thumbnail=\"/pelatihan-deep-learning-mathcitymap/foto-2.webp\" data-sub-html=\"\u003ch2\u003eMaterial Presentation by Unila PkM Team\u003c/h2\u003e\u003cp\u003eDeep Learning Material Presentation Session\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/pelatihan-deep-learning-mathcitymap/foto-2.webp' srcset=\"/pelatihan-deep-learning-mathcitymap/foto-2_hu_d562177984eb2f8c.webp 800w, /pelatihan-deep-learning-mathcitymap/foto-2_hu_740a2b286620c04f.webp 1200w, /pelatihan-deep-learning-mathcitymap/foto-2_hu_c41e2a9abab42688.webp 1600w\" alt=\"Deep Learning Material Presentation Session\" height=\"960\" width=\"1280\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eMaterial Presentation by Unila PkM Team\u003c/figcaption\u003e\n\u003c/figure\u003e\n\u003cem\u003ePhoto 2: The Unila PkM Team presenting conceptual material on integrating Deep Learning and utilizing MathCityMap features.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMathCityMap\u003c/strong\u003e (MCM)—a global platform developed by Goethe University Frankfurt—serves as an ideal tool to achieve this objective. The platform combines the concept of \u003cem\u003emath trails\u003c/em\u003e with Global Positioning System (GPS) technology and gamification, turning the surrounding environment into an open-air mathematics laboratory.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"training-structure-and-methodology\" class=\"headerLink\"\u003e\n \u003ca href=\"#training-structure-and-methodology\" class=\"header-mark\" aria-label=\"Direct link to section: Training Structure and Methodology\"\u003e\u003c/a\u003eTraining Structure and Methodology\u003c/h2\u003e\u003cp\u003eThe training was structured around a \u003cem\u003eWorkshop \u0026amp; Field-Based Practice\u003c/em\u003e scheme divided into two primary stages:\u003c/p\u003e\n\u003ch3 id=\"day-one-tuesday-july-21-2026-conceptualization--portal-design\" class=\"headerLink\"\u003e\n \u003ca href=\"#day-one-tuesday-july-21-2026-conceptualization--portal-design\" class=\"header-mark\" aria-label=\"Direct link to section: Day One (Tuesday, July 21, 2026): Conceptualization \u0026amp; Portal Design\"\u003e\u003c/a\u003eDay One (Tuesday, July 21, 2026): Conceptualization \u0026amp; Portal Design\u003c/h3\u003e\u003cp\u003eIn the first session, participants acquired theoretical insights on designing \u003cem\u003etasks\u003c/em\u003e that fulfill \u003cem\u003ehigher-order thinking skills\u003c/em\u003e (HOTS) criteria.\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/pelatihan-deep-learning-mathcitymap/foto-3.webp\" title=\"MathCityMap Web Platform\" data-thumbnail=\"/pelatihan-deep-learning-mathcitymap/foto-3.webp\" data-sub-html=\"\u003ch2\u003eMathCityMap Web Portal Interface\u003c/h2\u003e\u003cp\u003eMathCityMap Web Platform\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/pelatihan-deep-learning-mathcitymap/foto-3.webp' srcset=\"/pelatihan-deep-learning-mathcitymap/foto-3_hu_c4945ca7064607a7.webp 800w, /pelatihan-deep-learning-mathcitymap/foto-3_hu_41e5f3db69b17134.webp 1200w, /pelatihan-deep-learning-mathcitymap/foto-3_hu_ae58e30929d5b0e6.webp 1600w\" alt=\"MathCityMap Web Platform\" height=\"932\" width=\"1855\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eMathCityMap Web Portal Interface\u003c/figcaption\u003e\n\u003c/figure\u003e\n\u003cem\u003ePhoto 3: The MathCityMap web portal interface (mathcitymap.eu) used by teachers to upload tasks and plot GPS coordinates.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThrough the \u003ca href=\"https://mathcitymap.eu\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eMathCityMap\u003c/a\u003e web portal, participants were trained to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eDesign Contextual Tasks (\u003cem\u003eTask Design\u003c/em\u003e):\u003c/strong\u003e Create math problems whose solutions require direct data collection from real objects (e.g., calculating staircase inclination, school monument volume, or tile patterns).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDevelop Tiered Hints and Solutions:\u003c/strong\u003e Provide step-by-step guidance to foster independent student reasoning (\u003cem\u003escaffolding\u003c/em\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMap Trails:\u003c/strong\u003e Combine multiple \u003cem\u003etasks\u003c/em\u003e into one integrated exploration route (\u003cem\u003etrail\u003c/em\u003e) with precise GPS coordinates.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/pelatihan-deep-learning-mathcitymap/foto-4.webp\" title=\"Creating Tasks and Trails in MathCityMap\" data-thumbnail=\"/pelatihan-deep-learning-mathcitymap/foto-4.webp\" data-sub-html=\"\u003ch2\u003eCreating Tasks and Trails in MathCityMap\u003c/h2\u003e\u003cp\u003eCreating Tasks and Trails in MathCityMap\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/pelatihan-deep-learning-mathcitymap/foto-4.webp' srcset=\"/pelatihan-deep-learning-mathcitymap/foto-4_hu_3a3671ec95143c19.webp 800w, /pelatihan-deep-learning-mathcitymap/foto-4_hu_fda0c44addd0c458.webp 1200w, /pelatihan-deep-learning-mathcitymap/foto-4_hu_3f54dd28e66cc286.webp 1600w\" alt=\"Creating Tasks and Trails in MathCityMap\" height=\"890\" width=\"1863\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCreating Tasks and Trails in MathCityMap\u003c/figcaption\u003e\n\u003c/figure\u003e\n\u003cem\u003ePhoto 4: Creating tasks and trails within the MathCityMap application\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch3 id=\"day-two-wednesday-july-22-2026-field-simulation-validation-and-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#day-two-wednesday-july-22-2026-field-simulation-validation-and-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Day Two (Wednesday, July 22, 2026): Field Simulation, Validation, and Evaluation\"\u003e\u003c/a\u003eDay Two (Wednesday, July 22, 2026): Field Simulation, Validation, and Evaluation\u003c/h3\u003e\u003cp\u003eThe second day focused on practical implementation and trail testing within the premises of SMPN 4 Terbanggi Besar.\u003c/p\u003e\n\u003cp\u003ePractical activities included:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eObject Measurement and Modeling:\u003c/strong\u003e Teachers working in small teams conducted field explorations, measuring physical parameters of objects (such as angles, height, area, and volume) and crafting corresponding mathematical problems.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eApp-Based Trail Simulation:\u003c/strong\u003e Utilizing the MathCityMap mobile application to test GPS signal accuracy, instruction clarity, and automatic feedback mechanisms in real time.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePeer Review \u0026amp; Problem Calibration:\u003c/strong\u003e Reflective discussion among participants to ensure the tasks created were free of ambiguity and safe for student execution.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/pelatihan-deep-learning-mathcitymap/foto-5.webp\" title=\"MathCityMap Object Measurement\" data-thumbnail=\"/pelatihan-deep-learning-mathcitymap/foto-5.webp\" data-sub-html=\"\u003ch2\u003eObject Measurement by Workshop Participants\u003c/h2\u003e\u003cp\u003eMathCityMap Object Measurement\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/pelatihan-deep-learning-mathcitymap/foto-5.webp' srcset=\"/pelatihan-deep-learning-mathcitymap/foto-5_hu_79648a9acb7cf5d5.webp 800w, /pelatihan-deep-learning-mathcitymap/foto-5_hu_e4651cda77997e9d.webp 1200w, /pelatihan-deep-learning-mathcitymap/foto-5_hu_f9a8c23fcc1427e.webp 1600w\" alt=\"MathCityMap Object Measurement\" height=\"926\" width=\"1600\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eObject Measurement by Workshop Participants\u003c/figcaption\u003e\n\u003c/figure\u003e\n\u003cem\u003ePhoto 5: Photo of training participants performing measurements to answer questions provided in the MathCityMap trails.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/pelatihan-deep-learning-mathcitymap/foto-6.webp\" title=\"Training Activity Documentation\" data-thumbnail=\"/pelatihan-deep-learning-mathcitymap/foto-6.webp\" data-sub-html=\"\u003ch2\u003eAdditional Training Documentation\u003c/h2\u003e\u003cp\u003eTraining Activity Documentation\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/pelatihan-deep-learning-mathcitymap/foto-6.webp' srcset=\"/pelatihan-deep-learning-mathcitymap/foto-6_hu_b8375aae02bbe72d.webp 800w, /pelatihan-deep-learning-mathcitymap/foto-6_hu_b86af8e61f4a7069.webp 1200w, /pelatihan-deep-learning-mathcitymap/foto-6_hu_1d41b348037b522c.webp 1600w\" alt=\"Training Activity Documentation\" height=\"1280\" width=\"960\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eAdditional Training Documentation\u003c/figcaption\u003e\n\u003c/figure\u003e\n\u003cem\u003ePhoto 6: Interactive and enthusiastic atmosphere among mathematics teacher participants during the training sessions.\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"closing-and-future-outlook\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing-and-future-outlook\" class=\"header-mark\" aria-label=\"Direct link to section: Closing and Future Outlook\"\u003e\u003c/a\u003eClosing and Future Outlook\u003c/h2\u003e\u003cp\u003eThe entire event was officially closed by the Chairman of the Central Lampung Middle School Math MGMP, \u003cstrong\u003eAan Triono, S.Pd.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn his closing address, \u003cstrong\u003eAan Triono, S.Pd.\u003c/strong\u003e expressed deep appreciation for the dedication of the Unila PkM Team and the enthusiasm of all participants:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;May this collaboration continue and remain sustainable, enhancing the competence of mathematics teachers by implementing the MathCityMap application in their respective schools and compiling a regional math trail catalog accessible to all teachers and students.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"/pelatihan-deep-learning-mathcitymap/foto-7.webp\" title=\"Closing Ceremony Documentation\" data-thumbnail=\"/pelatihan-deep-learning-mathcitymap/foto-7.webp\" data-sub-html=\"\u003ch2\u003eClosing Ceremony Photo\u003c/h2\u003e\u003cp\u003eClosing Ceremony Documentation\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='/pelatihan-deep-learning-mathcitymap/foto-7.webp' srcset=\"/pelatihan-deep-learning-mathcitymap/foto-7_hu_55226ab90aea2a2a.webp 800w, /pelatihan-deep-learning-mathcitymap/foto-7_hu_6ab7a1da5a114a04.webp 1200w, /pelatihan-deep-learning-mathcitymap/foto-7_hu_44a925180b6cd2e3.webp 1600w\" alt=\"Closing Ceremony Documentation\" height=\"720\" width=\"1280\"\u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eClosing Ceremony Photo\u003c/figcaption\u003e\n\u003c/figure\u003e\n\u003cem\u003ePhoto 7: Photo of the training closing ceremony\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe synergy among university academics (Unila), the Department of Education, and the teacher community (MGMP) serves as an ideal model for accelerating regional education quality. Leveraging \u003cem\u003eMathCityMap\u003c/em\u003e with a \u003cem\u003eDeep Learning\u003c/em\u003e orientation proves to teachers that mathematics is not an isolated subject, but a living science that can be explored engagingly in our everyday surroundings.\u003c/p\u003e", "language": "en" }, { "title" : "Pascal's Triangle: Computation and Hexagonal Visualization in LaTeX", "date_published" : "2026-07-18T12:17:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pascal-triangle/", "url" : "https://www.aantriono.com/en/pascal-triangle/", "summary": "Pascal\u0026rsquo;s Triangle is a well-known triangular array of binomial coefficients in mathematics. Each number in the triangle is the sum of the two numbers directly above it. In addition to its usefulness in expanding algebraic expressions like (a+b)n(a+b)^n(a+b)n, this structure hosts many intriguing patterns.\nIn this article, we will build a visualization of Pascal\u0026rsquo;s Triangle in LaTeX using TikZ. Instead of writing the numbers manually, we will leverage TeX macro programming to compute the binomial coefficients dynamically and render them within colorful hexagonal (honeycomb) cells.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003e\u003cstrong\u003ePascal\u0026rsquo;s Triangle\u003c/strong\u003e is a well-known triangular array of binomial coefficients in mathematics. Each number in the triangle is the sum of the two numbers directly above it. In addition to its usefulness in expanding algebraic expressions like \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a+b)^n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6644em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, this structure hosts many intriguing patterns.\u003c/p\u003e\n\u003cp\u003eIn this article, we will build a visualization of Pascal\u0026rsquo;s Triangle in LaTeX using \u003cstrong\u003eTikZ\u003c/strong\u003e. Instead of writing the numbers manually, we will leverage TeX macro programming to compute the binomial coefficients dynamically and render them within colorful hexagonal (honeycomb) cells.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eGeometrically, a honeycomb layout (hexagonal tiling) is ideal for drawing Pascal\u0026rsquo;s Triangle since each hexagonal cell naturally borders two cells directly above it.\u003c/p\u003e\n\u003cp\u003eTo implement this efficiently in LaTeX, we require two key elements:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eComputational Algorithm\u003c/strong\u003e: An internal TeX macro to calculate the binomial coefficients \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e!\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e!\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e!\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{n}{k} = \\frac{n!}{k!(n-k)!}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7454em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.4001em;vertical-align:-0.52em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8801em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mclose mtight\"\u003e!\u003c/span\u003e\u003cspan class=\"mopen mtight\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mclose mtight\"\u003e)!\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003cspan class=\"mclose mtight\"\u003e!\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.52em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e dynamically without exceeding TeX compiler memory limits.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eGraphical Loops\u003c/strong\u003e: Nested TikZ loops (\u003ccode\u003e\\foreach\u003c/code\u003e) to draw the hexagons and place them at the correct coordinates according to their row and column values.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"complete-latex-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-latex-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete LaTeX Code\"\u003e\u003c/a\u003eComplete LaTeX Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code to generate the colored hexagonal Pascal\u0026rsquo;s Triangle:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Pascal triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: M.H. Ahmadi\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[border=10pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[dvipsnames]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eifthen\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\makeatletter\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Macro to compute binomial coefficients \\binomialCoefficient{n}{k}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcommand\\binomialCoefficient\u003c/span\u003e\u003cspan class=\"na\"\u003e[2]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% Store values \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@counta=#1\u003cspan class=\"c\"\u003e% n \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countb=#2\u003cspan class=\"c\"\u003e% k \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% Take advantage of symmetry if k \u0026gt; n - k \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countc=\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@counta\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\advance\\c\u003c/span\u003e@pgf@countc by-\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countb\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifnum\\c\u003c/span\u003e@pgf@countb\u0026gt;\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countc\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countb=\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countc\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\fi\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"c\"\u003e% Recursively compute the coefficients \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countc=1\u003cspan class=\"c\"\u003e% holds the result \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countd=0\u003cspan class=\"c\"\u003e% counter \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathloop\u003c/span\u003e\u003cspan class=\"c\"\u003e% formula: c -\u0026gt; c*(n-i)/(i+1) for i=0,...,k-1 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifnum\\c\u003c/span\u003e@pgf@countd\u0026lt;\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countb\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\multiply\\c\u003c/span\u003e@pgf@countc by\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@counta\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\advance\\c\u003c/span\u003e@pgf@counta by-1\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\advance\\c\u003c/span\u003e@pgf@countd by1\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\divide\\c\u003c/span\u003e@pgf@countc by\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e@pgf@countd\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\repeatpgfmathloop\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\the\\c\u003c/span\u003e@pgf@countc\u003cspan class=\"c\"\u003e% print result\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\makeatother\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newdimen\\R\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\R\u003c/span\u003e=.4cm \u003cspan class=\"c\"\u003e% Circumradius of the hexagon\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcommand\\mycolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egray\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[line width=.8pt] \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Loop for columns (k) from 0 to 12\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\k\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,12\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-60:\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(3)*\u003cspan class=\"k\"\u003e\\R*\\k\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e] \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathtruncatemacro\\ystart\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e12-\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Loop for rows (n)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,\u003cspan class=\"k\"\u003e\\ystart\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\pgfmathtruncatemacro\\newn\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\n\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Dynamic color selection based on columns/rows\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e=0\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\mycolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epink\u003cspan class=\"nb\"\u003e}}{}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e=1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\mycolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eyellow\u003cspan class=\"nb\"\u003e}}{}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e=2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\mycolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}}{}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e=3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\mycolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egreen\u003cspan class=\"nb\"\u003e}}{}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e=8 \u003cspan class=\"k\"\u003e\\AND\u003c/span\u003e \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e \u0026lt; 4\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\mycolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epurple\u003cspan class=\"nb\"\u003e}}{}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\ifthenelse\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\k\u003c/span\u003e=9 \u003cspan class=\"k\"\u003e\\AND\u003c/span\u003e \u003cspan class=\"k\"\u003e\\n\u003c/span\u003e = 3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\mycolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epurple\u003cspan class=\"nb\"\u003e}}{}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-120:\u003cspan class=\"nb\"\u003e{\u003c/span\u003esqrt(3)*\u003cspan class=\"k\"\u003e\\R*\\n\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e] \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Draw hexagons with top/bottom gradient colors\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[top color=\\mycolor!20,bottom color=\\mycolor!60]\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (30:\u003cspan class=\"k\"\u003e\\R\u003c/span\u003e) \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e90,150,...,330\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\R\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e -- cycle \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (90:0) node \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tiny\u003c/span\u003e \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\mathbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\binomialCoefficient\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\newn\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\k\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output of the code is as follows:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/pascal-triangle/featured-image.webp' srcset=\"/pascal-triangle/featured-image_hu_e20129d19fd8fe23.webp 800w, /pascal-triangle/featured-image_hu_41817c0da44953c6.webp 1200w, /pascal-triangle/featured-image_hu_ed9121709fe153c2.webp 1600w\" alt=\"Pascal\u0026rsquo;s Triangle Output\" height=\"720\" width=\"1280\"\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Explanation\"\u003e\u003c/a\u003eDetailed Code Explanation\u003c/h2\u003e\u003cp\u003eThe primary TeX and TikZ coding concepts used in the script are explained below:\u003c/p\u003e\n\u003ch3 id=\"1-low-level-tex-mathematical-computation-binomialcoefficient\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-low-level-tex-mathematical-computation-binomialcoefficient\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Low-Level TeX Mathematical Computation (\\binomialCoefficient)\"\u003e\u003c/a\u003e1. Low-Level TeX Mathematical Computation (\u003ccode\u003e\\binomialCoefficient\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eDue to TeX\u0026rsquo;s memory limits, calculating large factorials can easily cause buffer overflows. The calculation is optimized using low-level TeX counters (\u003ccode\u003e\\c@pgf@counta\u003c/code\u003e through \u003ccode\u003e\\c@pgf@countd\u003c/code\u003e) to perform fast and precise multiplication and division operations based on the multiplicative formula:\n\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmunderover\u003e\u003cmo\u003e∏\u003c/mo\u003e\u003cmrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/munderover\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{n}{k} = \\prod_{i=1}^{k} \\frac{n-i+1}{i}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.1138em;vertical-align:-1.2777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop op-limits\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.8361em;\"\u003e\u003cspan style=\"top:-1.8723em;margin-left:0em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.05em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003cspan class=\"mrel mtight\"\u003e=\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.05em;\"\u003e\u003c/span\u003e\u003cspan\u003e\u003cspan class=\"mop op-symbol large-op\"\u003e∏\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3em;margin-left:0em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.05em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2777em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3365em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\nIt also implements the symmetry identity \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{n}{k} = \\binom{n}{n-k}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7454em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2533em;vertical-align:-0.4033em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7454em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4033em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to cut down the iteration loop steps by half, speeding up the overall document compilation time.\u003c/p\u003e\n\u003ch3 id=\"2-drawing-regular-hexagons-using-polar-coordinates\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-drawing-regular-hexagons-using-polar-coordinates\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Drawing Regular Hexagons using Polar Coordinates\"\u003e\u003c/a\u003e2. Drawing Regular Hexagons using Polar Coordinates\u003c/h3\u003e\u003cp\u003eRegular hexagons are drawn dynamically using TikZ loops and polar coordinates:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e(30:\u003cspan class=\"k\"\u003e\\R\u003c/span\u003e) \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e90,150,...,330\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e:\u003cspan class=\"k\"\u003e\\R\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e -- cycle\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis starts drawing the path at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e30^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eR\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then creates line segments pointing to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e150\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e210\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e270\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e330\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^\\circ, 150^\\circ, 210^\\circ, 270^\\circ, 330^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8686em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e27\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e33\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and finally closes the path automatically (\u003ccode\u003e-- cycle\u003c/code\u003e) to form a perfect equilateral hexagon.\u003c/p\u003e\n\u003ch3 id=\"3-coordinate-vector-offsets-scope-and-shift\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-coordinate-vector-offsets-scope-and-shift\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Coordinate Vector Offsets (scope and shift)\"\u003e\u003c/a\u003e3. Coordinate Vector Offsets (\u003ccode\u003escope\u003c/code\u003e and \u003ccode\u003eshift\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eTo stack the hexagons into a triangular arrangement, we use nested \u003ccode\u003escope\u003c/code\u003e shifting coordinates:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThe column index is shifted along the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-60^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7574em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e vector by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{3} \\cdot R \\cdot k\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\n\u003ccode\u003eshift={(-60:{sqrt(3)*\\R*\\k})}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eThe row index is shifted along the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e120\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-120^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7574em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e vector by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{3} \\cdot R \\cdot n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\n\u003ccode\u003eshift={(-120:{sqrt(3)*\\R*\\n})}\u003c/code\u003e\nCombining these directional offsets correctly maps each cell to its corresponding slot in the symmetric triangle.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"4-dynamic-coloring-ifthenelse\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-dynamic-coloring-ifthenelse\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Dynamic Coloring (\\ifthenelse)\"\u003e\u003c/a\u003e4. Dynamic Coloring (\u003ccode\u003e\\ifthenelse\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eThe \u003ccode\u003eifthen\u003c/code\u003e package is used to apply different background gradients based on the column index:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eColumn 0 is shaded \u003ccode\u003epink\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eColumn 1 is shaded \u003ccode\u003eyellow\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eColumn 2 is shaded \u003ccode\u003eblue\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eColumn 3 is shaded \u003ccode\u003egreen\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eSpecific cells in the bottom-middle are shaded \u003ccode\u003epurple\u003c/code\u003e to highlight mathematical structures inside the triangle.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"how-to-compile-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-compile-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: How to Compile the Document\"\u003e\u003c/a\u003eHow to Compile the Document\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eEnsure a LaTeX distribution (such as TeX Live, MiKTeX, or Overleaf) is installed.\u003c/li\u003e\n\u003cli\u003eCopy the code above and save it as \u003ccode\u003epascal-triangle.tex\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eCompile using the \u003cstrong\u003epdfLaTeX\u003c/strong\u003e or \u003cstrong\u003eXeLaTeX\u003c/strong\u003e engine.\u003c/li\u003e\n\u003cli\u003eThe output page will be cropped to the size of the drawing with a 10pt border due to the \u003ccode\u003estandalone\u003c/code\u003e document class configuration.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eCalculating and drawing Pascal\u0026rsquo;s Triangle dynamically in TikZ LaTeX shows that LaTeX is more than a typesetting system—it is a complete, powerful programming environment. This technique is highly effective for designing clean, mathematically sound teaching materials and graphics without manual calculation. Happy TeXing!\u003c/p\u003e", "language": "en" }, { "title" : "Sri Yantra: Drawing Sacred Geometry in LaTeX", "date_published" : "2026-07-18T12:13:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sri-yantra/", "url" : "https://www.aantriono.com/en/sri-yantra/", "summary": "The Sri Yantra (also known as Sri Chakra) is one of the most mathematically complex geometric diagrams from ancient India, dating back to at least the 7th-8th century A.D. Its visual structure symbolizes the harmony of the cosmos and the unity of masculine (Shiva) and feminine (Shakti) energies.\nDrawing the Sri Yantra manually is a challenging feat, as it requires highly precise coordinates to ensure all intersecting lines meet exactly at their cross points. In this article, we will explore how to construct a mathematically precise Sri Yantra diagram using TikZ in LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThe \u003cstrong\u003eSri Yantra\u003c/strong\u003e (also known as Sri Chakra) is one of the most mathematically complex geometric diagrams from ancient India, dating back to at least the 7th-8th century A.D. Its visual structure symbolizes the harmony of the cosmos and the unity of masculine (Shiva) and feminine (Shakti) energies.\u003c/p\u003e\n\u003cp\u003eDrawing the Sri Yantra manually is a challenging feat, as it requires highly precise coordinates to ensure all intersecting lines meet exactly at their cross points. In this article, we will explore how to construct a mathematically precise Sri Yantra diagram using \u003cstrong\u003eTikZ\u003c/strong\u003e in LaTeX.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eGeometrically, the Sri Yantra is formed by the intersection of nine principal triangles (4 pointing upward representing the masculine Shiva energy, and 5 pointing downward representing the feminine Shakti energy).\u003c/p\u003e\n\u003cp\u003eThe intersection of these nine triangles creates a central matrix of 43 smaller triangles, surrounding a single central dot called the \u003cstrong\u003eBindu\u003c/strong\u003e. The entire structure is encircled by lotus petal rings (8 petals in the inner ring and 16 petals in the outer ring), enclosed by a temple gateway frame called the \u003cstrong\u003eBhupura\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eDue to its high complexity, even a tiny coordinate offset will create gaps or empty polygons at line intersection points. The code below uses verified coordinates refined through numerical optimization processes to ensure optimal precision.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"complete-latex-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-latex-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete LaTeX Code\"\u003e\u003c/a\u003eComplete LaTeX Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code using the TikZ library (\u003ccode\u003ebackgrounds\u003c/code\u003e and \u003ccode\u003emath\u003c/code\u003e) to draw the Sri Yantra:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz,border=2mm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebackgrounds,math\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfsetlayers\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus3,minus2,minus1,background,main\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Base parameters\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\cx\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e150\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\cy\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e150\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\scale\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e0.025\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\def\\triangleData\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e53.65669559977147\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e123.20508075688774\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e250\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e246.34330440022853\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% D1 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e52.984011026495736\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e174.24660560764943\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e50\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e247.01598897350425\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% U1 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e98.71823312733801\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e220.03828947357886\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e123.20508075688774\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e201.281766872662\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% U3 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e78.26467997914015\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e197.92315674002487\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e78.10499177949904\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e221.73532002085986\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% U2 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e90.4856922951427\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e78.10499177949904\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e160.66014976539617\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e209.51430770485734\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% D3 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e80.98384838952128\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e103.12199145016105\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e220.03828947357886\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e219.0161516104787\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% D2 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e114.9488500600036\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e160.66014976539617\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e103.12199145016105\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e185.0511499399964\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% U4 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e116.35142605010424\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e134.30757626706648\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e197.92315674002487\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e183.64857394989576\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% D4 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e124.61190803072795\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e144.79777263138968\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e174.24660560764943\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e175.38809196927207\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% D5 \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Function to transform and draw triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Usage: \\drawtriangle{x1}{y1}{x2}{y2}{x3}{y3} \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\drawtriangle\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[6]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick]\u003c/span\u003e (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e((#1-\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-(#2-\u003cspan class=\"k\"\u003e\\cy\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e) -- (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e((#3-\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-(#4-\u003cspan class=\"k\"\u003e\\cy\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e) -- (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e((#5-\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-(#6-\u003cspan class=\"k\"\u003e\\cy\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e) -- cycle;\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Macro to draw a triangle from array data\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Arguments: left_x, top/base_y, center_y, right_x, type \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\drawTriangleFromData\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[5]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ifnum\u003c/span\u003e#5=0 \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Downward triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\drawtriangle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#4\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\else\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% Upward triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\drawtriangle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#4\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\fi\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Recursive macro to process triangle data\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Stops when it encounters \\relax as the first argument \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\def\\processTriangleData\u003c/span\u003e#1#2#3#4#5\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ifx\\relax\u003c/span\u003e#1\u003cspan class=\"k\"\u003e\\relax\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e% End of data reached \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\else\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\drawTriangleFromData\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#4\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\expandafter\\processTriangleData\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fi\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Outer circle \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick, fill=white!30!white]\u003c/span\u003e (0,0) circle (2.5cm); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Center bindu \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=red]\u003c/span\u003e (0,0) circle (0.1cm); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Draw all triangles from triangleData array \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\expandafter\\processTriangleData\\triangleData\\relax\\relax\\relax\\relax\\relax\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Step 11: Outer Lotus Petals (8 petals)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[on background layer] \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [fill=yellow] circle (3cm); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e [evaluate=\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e as \u003cspan class=\"k\"\u003e\\start\u003c/span\u003e using 22.5+45*\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e] in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,7\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=green!30!black]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e:2.5) .. controls (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:2.85) and (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-17.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:2.75) .. (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-22.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3) .. controls (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-27.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:2.75) and (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-40\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:2.85) .. (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-45\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:2.5) arc [start angle=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-45\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, delta angle=45, radius=2.5cm]--cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Step 12: Outer Lotus Petals (16 petals)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfonlayer\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eminus1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [fill=orange] circle (3.5cm); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e [evaluate=\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e as \u003cspan class=\"k\"\u003e\\start\u003c/span\u003e using 11.25+22.5*\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e] in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,15\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=blue!30]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e:3) .. controls (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-2.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3.35) and (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-8.75\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3.25) .. (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-11.25\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3.5) .. controls (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-13.75\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3.25) and (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-20\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3.35).. (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-22.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3) arc [start angle=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-11.25\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, delta angle=22.5, radius=3.5cm]--cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfonlayer\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Step 13: Concentric Circles\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfonlayer\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eminus2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [fill=yellow!39] circle (4cm); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [line width=2mm, blue!70!black] circle (3.75cm); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfonlayer\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Step 14: Bhupura (Square Temple Gates)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfonlayer\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eminus3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [fill=black, minimum size=8cm] (b) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [fill=black, minimum width=9.25cm, minimum height=3cm] (bh) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [fill=black, minimum height=9.25cm, minimum width=3cm] (bv) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [fill=black, minimum height=6mm, minimum width=6cm] at (bv.north) (bn) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [fill=black, minimum height=6mm, minimum width=6cm] at (bv.south) (bs) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [fill=black, minimum width=6mm, minimum height=6cm] at (bh.west) (bw) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [fill=black, minimum width=6mm, minimum height=6cm] at (bh.east) (be) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [red!60!black,line width=1mm]\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e(be.south west)-|(be.north east)-|(bh.north-|be.west)-|(b.north east)-|(bv.east|-bn.south)-|(bn.north east)-|(bn.south west)-|(b.north-|bv.west)-|(b.west|-bh.north)-|(bw.north east)-|(bw.south west)-|(bw.east|-bh.south)-|(b.south west)-|(bv.west|-bs.north)-|(bs.south west)-|(bs.north east)-|(bv.east|-b.south)-|(b.east|-bh.south)-|cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfonlayer\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output of the code is as follows:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/sri-yantra/featured-image.webp' srcset=\"/sri-yantra/featured-image_hu_a1797a02eca25886.webp 800w, /sri-yantra/featured-image_hu_47de75282bb787f0.webp 1200w, /sri-yantra/featured-image_hu_5e550703f9455cc6.webp 1600w\" alt=\"Sri Yantra Output\" height=\"720\" width=\"1280\"\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Explanation\"\u003e\u003c/a\u003eDetailed Code Explanation\u003c/h2\u003e\u003cp\u003eLet\u0026rsquo;s dissect the main components of the LaTeX code:\u003c/p\u003e\n\u003ch3 id=\"1-drawing-layers-pgf-layers\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-drawing-layers-pgf-layers\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Drawing Layers (PGF Layers)\"\u003e\u003c/a\u003e1. Drawing Layers (PGF Layers)\u003c/h3\u003e\u003cp\u003eTo render outer decorative details (such as the lotus petals and the Bhupura gates) behind the main intersecting triangles, we declare custom layers:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfdeclarelayer\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfsetlayers\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminus3,minus2,minus1,background,main\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThese layers are stacked from bottom (\u003ccode\u003eminus3\u003c/code\u003e) to top (\u003ccode\u003emain\u003c/code\u003e), ensuring that background patterns do not overlap or obscure the central vector lines of the triangles.\u003c/p\u003e\n\u003ch3 id=\"2-coordinate-system-mapping\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-coordinate-system-mapping\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Coordinate System Mapping\"\u003e\u003c/a\u003e2. Coordinate System Mapping\u003c/h3\u003e\u003cp\u003eThe SVG coordinates of the triangles were originally mapped with a center at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e150\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e150\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(150, 150)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e150\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e150\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and a radius of 100. In TikZ, we want a coordinate space centered at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0, 0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with a radius of 2.5cm. We define the transformation:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eScaling factor: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2.5\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e/\u003c/mi\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.025\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2.5 / 100 = 0.025\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2.5/100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.025\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (\u003ccode\u003e\\scale\u003c/code\u003e)\u003c/li\u003e\n\u003cli\u003eX-axis transformation: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmtext\u003eoriginal\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003escale\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_{\\text{original}} - cx) \\times \\text{scale}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003eoriginal\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003escale\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eY-axis transformation: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmtext\u003eoriginal\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003escale\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-(y_{\\text{original}} - cy) \\times \\text{scale}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003eoriginal\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ecy\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003escale\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\nWe package this into the \u003ccode\u003e\\drawtriangle\u003c/code\u003e macro:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\drawtriangle\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[6]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick]\u003c/span\u003e (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e((#1-\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-(#2-\u003cspan class=\"k\"\u003e\\cy\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e) -- (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e((#3-\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-(#4-\u003cspan class=\"k\"\u003e\\cy\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e) -- (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e((#5-\u003cspan class=\"k\"\u003e\\cx\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(-(#6-\u003cspan class=\"k\"\u003e\\cy\u003c/span\u003e)*\u003cspan class=\"k\"\u003e\\scale\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e) -- cycle;\u003cspan class=\"c\"\u003e% \n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"3-recursive-macro-parsing-processtriangledata\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-recursive-macro-parsing-processtriangledata\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Recursive Macro Parsing (\\processTriangleData)\"\u003e\u003c/a\u003e3. Recursive Macro Parsing (\u003ccode\u003e\\processTriangleData\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eInstead of issuing individual \u003ccode\u003e\\draw\u003c/code\u003e commands for nine different triangles, the vertices are structured into an array-like variable \u003ccode\u003e\\triangleData\u003c/code\u003e with the structure:\n\u003ccode\u003e{left_x}{top/base_y}{center_y}{right_x}{type}\u003c/code\u003e\nThe recursive macro \u003ccode\u003e\\processTriangleData\u003c/code\u003e reads these parameters sequentially and automatically terminates when it encounters the \u003ccode\u003e\\relax\u003c/code\u003e keyword. This makes the code highly modular and compact.\u003c/p\u003e\n\u003ch3 id=\"4-bezier-curves-for-lotus-petals\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-bezier-curves-for-lotus-petals\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Bezier Curves for Lotus Petals\"\u003e\u003c/a\u003e4. Bezier Curves for Lotus Petals\u003c/h3\u003e\u003cp\u003eThe lotus petals are drawn automatically using loops (\u003ccode\u003e\\foreach\u003c/code\u003e) combined with Bezier control points:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e [evaluate=\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e as \u003cspan class=\"k\"\u003e\\start\u003c/span\u003e using 22.5+45*\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e] in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0,...,7\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=green!30!black]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e:2.5) .. controls (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:2.85) and (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-17.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:2.75) .. (\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\start\u003c/span\u003e-22.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e:3) ...\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis algorithmically draws symmetric, curved petals at specified angular intervals around the concentric circles.\u003c/p\u003e\n\u003ch3 id=\"5-bhupura-temple-gate\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-bhupura-temple-gate\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Bhupura Temple Gate\"\u003e\u003c/a\u003e5. Bhupura Temple Gate\u003c/h3\u003e\u003cp\u003eThe Bhupura (outer border structure) is formed by stacking multiple rectangular nodes horizontally and vertically to build the signature four-sided gate contour, which is then outlined with a single, continuous path \u003ccode\u003e\\draw\u003c/code\u003e tracing the outer boundary of the node coordinates.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"how-to-compile-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-compile-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: How to Compile the Document\"\u003e\u003c/a\u003eHow to Compile the Document\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eEnsure you have a LaTeX editor/distribution (like TeX Live, MiKTeX, or Overleaf) ready.\u003c/li\u003e\n\u003cli\u003eSave the code above into a file named \u003ccode\u003esri-yantra.tex\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eCompile using the \u003cstrong\u003epdfLaTeX\u003c/strong\u003e or \u003cstrong\u003eXeLaTeX\u003c/strong\u003e engine.\u003c/li\u003e\n\u003cli\u003eThe \u003ccode\u003estandalone\u003c/code\u003e document class with \u003ccode\u003eborder=2mm\u003c/code\u003e will crop the output PDF precisely to the drawing\u0026rsquo;s dimensions, leaving no empty space.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe Sri Yantra is a prime example of how ancient iconography embeds profound geometric complexity. With the help of TeX macro programming and TikZ vector capabilities in LaTeX, this sacred geometry can be generated with absolute mathematical accuracy, presenting an intriguing study in math, art, and graphics programming. Happy TeXing!\u003c/p\u003e", "language": "en" }, { "title" : "Perigal's Dissection: A Geometric Proof of the Pythagorean Theorem in LaTeX", "date_published" : "2026-07-18T12:05:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/perigals-dissection/", "url" : "https://www.aantriono.com/en/perigals-dissection/", "summary": "The Pythagorean theorem is one of the most famous fundamental principles in mathematics. There are many ways to prove it, both algebraically and geometrically. One of the most elegant and visual geometric proofs is Perigal\u0026rsquo;s Dissection, discovered by Henry Perigal in 1873.\nIn this article, we will learn how to create a visualization of Perigal\u0026rsquo;s Dissection using the TikZ package in LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThe Pythagorean theorem is one of the most famous fundamental principles in mathematics. There are many ways to prove it, both algebraically and geometrically. One of the most elegant and visual geometric proofs is \u003cstrong\u003ePerigal\u0026rsquo;s Dissection\u003c/strong\u003e, discovered by Henry Perigal in 1873.\u003c/p\u003e\n\u003cp\u003eIn this article, we will learn how to create a visualization of Perigal\u0026rsquo;s Dissection using the \u003cstrong\u003eTikZ\u003c/strong\u003e package in LaTeX.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003ePerigal\u0026rsquo;s dissection is a method for proving the Pythagorean theorem (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea^2 + b^2 = c^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) by cutting (dissecting) the square constructed on the longer leg of a right-angled triangle into four congruent parts.\u003c/p\u003e\n\u003cp\u003eThese cuts are made by drawing two perpendicular lines passing through the center of the square, where one of the lines is parallel to the hypotenuse of the triangle.\u003c/p\u003e\n\u003cp\u003eThese four parts, along with the intact square of the shorter leg, can be rearranged perfectly to cover the entire area of the square constructed on the hypotenuse.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"complete-latex-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-latex-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete LaTeX Code\"\u003e\u003c/a\u003eComplete LaTeX Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code using the TikZ library (\u003ccode\u003eintersections\u003c/code\u003e and \u003ccode\u003emath\u003c/code\u003e) to generate the Perigal\u0026rsquo;s dissection diagram precisely:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" 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MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 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72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eintersections\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emath\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[active,tightpage]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epreview\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\PreviewEnvironment\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\setlength\\PreviewBorder\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e0.125pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% File name: perigals-dissection.tex\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Description: A geometric proof of the Pythagorean theorem using Perigal\u0026#39;s dissection.\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Author: Efraín Soto Apolinar.\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[scale=0.75] \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzmath\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efunction rotacionx(\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003ereturn \u003cspan class=\"k\"\u003e\\x*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) + \u003cspan class=\"k\"\u003e\\y*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzmath\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efunction rotaciony(\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003ereturn -\u003cspan class=\"k\"\u003e\\x*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) + \u003cspan class=\"k\"\u003e\\y*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\dx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.25\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Vertices of the right triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (A) at (\u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\By\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (B) at (\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\By\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (C) at (\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Lengths of the sides\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\lAB\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt((\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e)*(\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e) + (\u003cspan class=\"k\"\u003e\\By\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e)*(\u003cspan class=\"k\"\u003e\\By\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e))\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\lAC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt((\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e)*(\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e) + (\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e)*(\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e))\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt((\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e)*(\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e) + (\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\By\u003c/span\u003e)*(\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\By\u003c/span\u003e))\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Slopes of the sides of the triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\mAC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e(\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e)/(\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\mAB\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e(\u003cspan class=\"k\"\u003e\\By\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e)/(\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\angA\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eatan(\u003cspan class=\"k\"\u003e\\mAC\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\angB\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e90.0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\angC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e90.0-\u003cspan class=\"k\"\u003e\\angA\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\angAH\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e45+\u003cspan class=\"k\"\u003e\\angA\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Coordinates of the vertices of the squares on the sides\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Dx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Dy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\lAB\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (D) at (\u003cspan class=\"k\"\u003e\\Dx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Dy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ex\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ey\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\lAB\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (E) at (\u003cspan class=\"k\"\u003e\\Ex\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ey\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Fx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Fy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (F) at (\u003cspan class=\"k\"\u003e\\Fx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Fy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Gx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Gy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (G) at (\u003cspan class=\"k\"\u003e\\Gx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Gy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Hx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt(2.0)*\u003cspan class=\"k\"\u003e\\lAC*\u003c/span\u003ecos(45+\u003cspan class=\"k\"\u003e\\angA\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Hy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt(2.0)*\u003cspan class=\"k\"\u003e\\lAC*\u003c/span\u003esin(45+\u003cspan class=\"k\"\u003e\\angA\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (H) at (\u003cspan class=\"k\"\u003e\\Hx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Hy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ix\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003erotacionx(\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e,-90)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Iy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003erotaciony(\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e,-90)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (I) at (\u003cspan class=\"k\"\u003e\\Ix\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Iy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Coordinates of the centroid of the square upon the hypotenuse\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Mx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.5*(\u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\Hx\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\My\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.5*(\u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e+\u003cspan class=\"k\"\u003e\\Hy\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (M) at (\u003cspan class=\"k\"\u003e\\Mx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\My\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Points upon the hypotenuse to draw the dissection\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Jx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Mx\u003c/span\u003e+0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Jy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\My\u003c/span\u003e+0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (J) at (\u003cspan class=\"k\"\u003e\\Jx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Jy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Kx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Mx\u003c/span\u003e-0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ky\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\My\u003c/span\u003e+0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (K) at (\u003cspan class=\"k\"\u003e\\Kx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ky\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Lx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Mx\u003c/span\u003e-0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ly\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\My\u003c/span\u003e-0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (L) at (\u003cspan class=\"k\"\u003e\\Lx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ly\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Nx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Mx\u003c/span\u003e+0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ny\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\My\u003c/span\u003e-0.5*\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (N) at (\u003cspan class=\"k\"\u003e\\Nx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ny\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Length for dissection\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\mCH\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e(\u003cspan class=\"k\"\u003e\\Hy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e)/(\u003cspan class=\"k\"\u003e\\Hx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\xp\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e+(\u003cspan class=\"k\"\u003e\\Jy\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e)/\u003cspan class=\"k\"\u003e\\mCH\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\yp\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Jy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ld\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esqrt((\u003cspan class=\"k\"\u003e\\Jx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\xp\u003c/span\u003e)*(\u003cspan class=\"k\"\u003e\\Jx\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\xp\u003c/span\u003e) + (\u003cspan class=\"k\"\u003e\\Jy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\yp\u003c/span\u003e) * (\u003cspan class=\"k\"\u003e\\Jy\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\yp\u003c/span\u003e))\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Vertices of the dissection upon the longer leg (AB)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Cunox\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Cunoy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\By\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\ld\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Cuno) at (\u003cspan class=\"k\"\u003e\\Cunox\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Cunoy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Cdosx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\ld\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Cdosy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Cdos) at (\u003cspan class=\"k\"\u003e\\Cdosx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Cdosy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ctresx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Dx\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ctresy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Dy\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\ld\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Ctres) at (\u003cspan class=\"k\"\u003e\\Ctresx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ctresy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ccuatrox\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ex\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\ld\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ccuatroy\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Ey\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Ccuatro) at (\u003cspan class=\"k\"\u003e\\Ccuatrox\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ccuatroy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=UT]\u003c/span\u003e (Cuno) -- (Ctres); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=DC]\u003c/span\u003e (Cdos) -- (Ccuatro); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name intersections={of=UT and DC}]\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Q) at (intersection-1); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ejex\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003emax(\u003cspan class=\"k\"\u003e\\Ax\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Cx\u003c/span\u003e) + 1.0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfmathsetmacro\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ejey\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003emax(\u003cspan class=\"k\"\u003e\\Ay\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\By\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Cy\u003c/span\u003e) + 1.0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% The right triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\dx\u003c/span\u003e,0) -- (\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\dx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\dx\u003c/span\u003e) -- (\u003cspan class=\"k\"\u003e\\Bx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\dx\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue,thick]\u003c/span\u003e (A) -- (B) -- (C) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e (A) circle (1.0pt) node[below left]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eA\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e (B) circle (1.0pt) node[below right]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eB\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e (C) circle (1.0pt) node[above right]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eC\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Squares on the sides of the right triangle\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,thick]\u003c/span\u003e (0,0) rectangle (\u003cspan class=\"k\"\u003e\\lAB\u003c/span\u003e,-\u003cspan class=\"k\"\u003e\\lAB\u003c/span\u003e); \u003cspan class=\"c\"\u003e% Side AB (longer leg)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,thick,rotate=\\angA]\u003c/span\u003e (0,0) rectangle (\u003cspan class=\"k\"\u003e\\lAC\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\lAC\u003c/span\u003e); \u003cspan class=\"c\"\u003e% Side AC (hypotenuse)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue,thick,fill=cyan!15,shift={(\\Bx,\\By)}]\u003c/span\u003e (0,0) rectangle (\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\lBC\u003c/span\u003e); \u003cspan class=\"c\"\u003e% Side BC (shorter leg)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Rectangle of dissection\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue,thick,fill=cyan!15]\u003c/span\u003e (J) -- (K) -- (L) -- (N) -- (J); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Compute the intersections to draw the dissection\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Jp) at (\u003cspan class=\"k\"\u003e\\Jx\u003c/span\u003e+2.0,\u003cspan class=\"k\"\u003e\\Jy\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=CH]\u003c/span\u003e (C) -- (H); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=JJp]\u003c/span\u003e (J) -- (Jp); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name intersections={of=CH and JJp}]\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Huno) at (intersection-1); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Np) at (\u003cspan class=\"k\"\u003e\\Nx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ny\u003c/span\u003e-2.0); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=AC]\u003c/span\u003e (A) -- (C); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=NNp]\u003c/span\u003e (N) -- (Np); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name intersections={of=AC and NNp}]\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Hdos) at (intersection-1); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Lp) at (\u003cspan class=\"k\"\u003e\\Lx\u003c/span\u003e-2.0,\u003cspan class=\"k\"\u003e\\Ly\u003c/span\u003e); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=AI]\u003c/span\u003e (A) -- (I); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=LLp]\u003c/span\u003e (L) -- (Lp); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name intersections={of=AI and LLp}]\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Htres) at (intersection-1); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Kp) at (\u003cspan class=\"k\"\u003e\\Kx\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\Ky\u003c/span\u003e+2.0); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=HI]\u003c/span\u003e (H) -- (I); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=KKp]\u003c/span\u003e (K) -- (Kp); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name intersections={of=HI and KKp}]\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Hcuatro) at (intersection-1); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Disection upon the hypotenuse\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=yellow!35,opacity=0.5]\u003c/span\u003e (C) -- (Huno) -- (J) -- (Hdos) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=gray!35,opacity=0.5]\u003c/span\u003e (A) -- (Hdos) -- (N) -- (Htres) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=pink!35,opacity=0.5]\u003c/span\u003e (I) -- (Htres) -- (L) -- (Hcuatro) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=green!35,opacity=0.5]\u003c/span\u003e (H) -- (Huno) -- (K) -- (Hcuatro) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% Disection upon the longest leg (AB)\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=gray!35,opacity=0.5]\u003c/span\u003e (Q) -- (Cuno) -- (B) -- (Cdos) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=yellow!35,opacity=0.5]\u003c/span\u003e (Q) -- (Cdos) -- (A) -- (Ctres) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=green!35,opacity=0.5]\u003c/span\u003e (Q) -- (Ctres) -- (D) -- (Ccuatro) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[cyan,fill=pink!35,opacity=0.5]\u003c/span\u003e (Q) -- (Ccuatro) -- (E) -- (Cuno) -- cycle; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output of the code is as follows:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/perigals-dissection/featured-image.webp' srcset=\"/perigals-dissection/featured-image_hu_7531b74d2d457dbe.webp 800w, /perigals-dissection/featured-image_hu_2481dc618262f626.webp 1200w, /perigals-dissection/featured-image_hu_56b268caaa319f74.webp 1600w\" alt=\"Perigal\u0026rsquo;s Dissection Output\" height=\"720\" width=\"1280\"\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Explanation\"\u003e\u003c/a\u003eDetailed Code Explanation\u003c/h2\u003e\u003cp\u003eLet\u0026rsquo;s analyze the mathematical concepts and main TikZ commands used in the code above:\u003c/p\u003e\n\u003ch3 id=\"1-custom-coordinate-rotation-tikzmath\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-custom-coordinate-rotation-tikzmath\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Custom Coordinate Rotation (\\tikzmath)\"\u003e\u003c/a\u003e1. Custom Coordinate Rotation (\u003ccode\u003e\\tikzmath\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eTo calculate geometric rotation positions, custom functions were defined using the \u003ccode\u003emath\u003c/code\u003e library:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzmath\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efunction rotacionx(\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003ereturn \u003cspan class=\"k\"\u003e\\x*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) + \u003cspan class=\"k\"\u003e\\y*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tikzmath\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efunction rotaciony(\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003ereturn -\u003cspan class=\"k\"\u003e\\x*\u003c/span\u003esin(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e) + \u003cspan class=\"k\"\u003e\\y*\u003c/span\u003ecos(\u003cspan class=\"k\"\u003e\\angulo\u003c/span\u003e);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe \u003ccode\u003erotacionx\u003c/code\u003e and \u003ccode\u003erotaciony\u003c/code\u003e functions calculate the new \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e coordinates after rotating a point by \u003ccode\u003e\\angulo\u003c/code\u003e degrees. This is essential for precisely aligning the square on the inclined hypotenuse.\u003c/p\u003e\n\u003ch3 id=\"2-parameterization-using-pgf-pgfmathsetmacro\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-parameterization-using-pgf-pgfmathsetmacro\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Parameterization Using PGF (\\pgfmathsetmacro)\"\u003e\u003c/a\u003e2. Parameterization Using PGF (\u003ccode\u003e\\pgfmathsetmacro\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eGeometric constants such as the side lengths of the right triangle (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB = 4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBC = 3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, hypotenuse \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAC = 5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e), the slope angle of the hypotenuse (\u003ccode\u003e\\angA\u003c/code\u003e), and the center coordinates of the dissection are computed dynamically using \u003ccode\u003e\\pgfmathsetmacro\u003c/code\u003e. This ensures the diagram is precise and scalable.\u003c/p\u003e\n\u003ch3 id=\"3-path-intersections-intersections\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-path-intersections-intersections\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Path Intersections (intersections)\"\u003e\u003c/a\u003e3. Path Intersections (\u003ccode\u003eintersections\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eThe cutting lines of the dissection are computed using the \u003ccode\u003eintersections\u003c/code\u003e library. TikZ automatically detects the meeting point between two pathways:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=UT]\u003c/span\u003e (Cuno) -- (Ctres); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=DC]\u003c/span\u003e (Cdos) -- (Ccuatro); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name intersections={of=UT and DC}]\u003c/span\u003e; \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (Q) at (intersection-1);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis line detects the intersection point \u003ccode\u003eQ\u003c/code\u003e, which serves as the centroid of the dissection cuts on square \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e (the longer leg).\u003c/p\u003e\n\u003ch3 id=\"4-coloring-the-geometrical-pieces\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-coloring-the-geometrical-pieces\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Coloring the Geometrical Pieces\"\u003e\u003c/a\u003e4. Coloring the Geometrical Pieces\u003c/h3\u003e\u003cp\u003eEach dissection slice is colored with a different fill (\u003ccode\u003eyellow\u003c/code\u003e, \u003ccode\u003egray\u003c/code\u003e, \u003ccode\u003epink\u003c/code\u003e, \u003ccode\u003egreen\u003c/code\u003e) at half transparency (\u003ccode\u003eopacity=0.5\u003c/code\u003e) to help readers visually trace how the pieces from the bottom square (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) can be rearranged to fit perfectly into the square on the hypotenuse.\u003c/p\u003e\n\u003cp\u003eThe square on the shorter leg (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) is filled with light blue (\u003ccode\u003ecyan!15\u003c/code\u003e) and is positioned in the center of the hypotenuse square as the fifth compiling piece.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"how-to-compile-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-compile-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: How to Compile the Document\"\u003e\u003c/a\u003eHow to Compile the Document\u003c/h2\u003e\u003cp\u003eTo compile the LaTeX file:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eEnsure a LaTeX environment (such as TeX Live, MiKTeX, or Overleaf) is installed.\u003c/li\u003e\n\u003cli\u003eCopy the LaTeX code above into a new file named \u003ccode\u003eperigals-dissection.tex\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eCompile using \u003cstrong\u003epdfLaTeX\u003c/strong\u003e or \u003cstrong\u003eXeLaTeX\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eThe \u003ccode\u003epreview\u003c/code\u003e package will crop the output PDF page precisely to fit the boundaries of the TikZ picture.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003ePerigal\u0026rsquo;s dissection provides a visually striking proof of the Pythagorean theorem without requiring complex algebraic formulas. With the help of the \u003ccode\u003emath\u003c/code\u003e and \u003ccode\u003eintersections\u003c/code\u003e libraries in TikZ, we can create precise and elegant representations of classical geometry theorems. Happy TeXing!\u003c/p\u003e", "language": "en" }, { "title" : "Scholars: Heirs of the Prophets vs. Heirs of Desires", "date_published" : "2026-07-16T21:05:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/ulama-pewaris-nabi-dan-ulama-pewaris-nafsu/", "url" : "https://www.aantriono.com/en/ulama-pewaris-nabi-dan-ulama-pewaris-nafsu/", "summary": " IntroductionTrue scholars (ulama) occupy an exceptionally noble position in Islam. They are the heirs of the Prophets—not only inheriting the treasury of knowledge, but also carrying the heavy responsibility of calling people to Allah (dawah). Consequently, the actions and stances of true scholars will always emulate the footsteps of the prophets of the past, including in how they interact with rulers and those in authority.\nOn the other hand, history and contemporary reality also reveal a class of religious scholars who sacrifice their knowledge and religious principles for worldly interests, power, or popularity. This group is often termed scholars who inherit desires (ulama su\u0026rsquo;). Understanding the fundamental differences between these two is vital so that we do not mistakenly choose our role models.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eTrue scholars (\u003cem\u003eulama\u003c/em\u003e) occupy an exceptionally noble position in Islam. They are the heirs of the Prophets—not only inheriting the treasury of knowledge, but also carrying the heavy responsibility of calling people to Allah (\u003cem\u003edawah\u003c/em\u003e). Consequently, the actions and stances of true scholars will always emulate the footsteps of the prophets of the past, including in how they interact with rulers and those in authority.\u003c/p\u003e\n\u003cp\u003eOn the other hand, history and contemporary reality also reveal a class of religious scholars who sacrifice their knowledge and religious principles for worldly interests, power, or popularity. This group is often termed \u003cstrong\u003escholars who inherit desires\u003c/strong\u003e (\u003cem\u003eulama su\u0026rsquo;\u003c/em\u003e). Understanding the fundamental differences between these two is vital so that we do not mistakenly choose our role models.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"three-relational-paths-of-prophets-and-power-inherited-by-scholars\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-relational-paths-of-prophets-and-power-inherited-by-scholars\" class=\"header-mark\" aria-label=\"Direct link to section: Three Relational Paths of Prophets and Power Inherited by Scholars\"\u003e\u003c/a\u003eThree Relational Paths of Prophets and Power Inherited by Scholars\u003c/h2\u003e\u003cp\u003eIf we examine the stories of the prophets in the Qur\u0026rsquo;an, their interaction with political power falls into three main categories. This legacy is what true scholars inherit:\u003c/p\u003e\n\u003ch3 id=\"1-prophets-who-were-also-rulers\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-prophets-who-were-also-rulers\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Prophets Who Were Also Rulers\"\u003e\u003c/a\u003e1. Prophets Who Were Also Rulers\u003c/h3\u003e\u003cp\u003eProphet Daud (David) and Prophet Sulaiman (Solomon) \u0026lsquo;alaihimas salam are prime examples of prophets who held absolute power in their time. They utilized this authority solely to establish justice and implement the laws of Allah SWT.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eTheir Heirs:\u003c/strong\u003e Noble companions like the Rightly Guided Caliphs (Abu Bakar, Umar, Utsman, and Ali). They were both great scholars and heads of state who led Islam to its golden age without ever being seduced by the glitter of the throne.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-prophets-who-entered-governance-for-reform\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-prophets-who-entered-governance-for-reform\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Prophets Who Entered Governance for Reform\"\u003e\u003c/a\u003e2. Prophets Who Entered Governance for Reform\u003c/h3\u003e\u003cp\u003eProphet Yusuf (Joseph) \u0026lsquo;alaihissalam consciously requested an influential administrative position in the kingdom of Egypt to save the community from a prolonged famine while propagating monotheism (\u003cem\u003etawheed\u003c/em\u003e).\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eTheir Heirs:\u003c/strong\u003e Scholars who remain close to rulers, not to seek personal wealth or act as sycophants, but to offer sincere counsel and keep rulers on the path of truth. An example is Imam Malik. Despite his proximity to the Caliphs, he remained highly critical and was once physically lashed for issuing a fatwa that contradicted state policy.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-prophets-who-stood-firm-against-tyrannical-rulers\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-prophets-who-stood-firm-against-tyrannical-rulers\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Prophets Who Stood Firm Against Tyrannical Rulers\"\u003e\u003c/a\u003e3. Prophets Who Stood Firm Against Tyrannical Rulers\u003c/h3\u003e\u003cp\u003eThe \u003cem\u003eUlul Azmi\u003c/em\u003e prophets like Prophet Ibrahim (Abraham) who confronted Nimrod, Prophet Musa (Moses) who opposed Pharaoh, and Prophet Muhammad SAW who stood against the chieftains of Quraish.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eTheir Heirs:\u003c/strong\u003e Scholars who stand firm against the injustice of regimes. Examples include Sa\u0026rsquo;id bin Jubayr, who was executed for criticizing the dictatorial governor Al-Hajjaj bin Yusuf, and Imam Ahmad bin Hanbal, who was imprisoned and tortured for years for defending the correct creed against the state-sponsored deviant theology.\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"characteristics-heirs-of-the-prophets-vs-heirs-of-desires\" class=\"headerLink\"\u003e\n \u003ca href=\"#characteristics-heirs-of-the-prophets-vs-heirs-of-desires\" class=\"header-mark\" aria-label=\"Direct link to section: Characteristics: Heirs of the Prophets vs. Heirs of Desires\"\u003e\u003c/a\u003eCharacteristics: Heirs of the Prophets vs. Heirs of Desires\u003c/h2\u003e\u003cp\u003eIn today\u0026rsquo;s secular age, religion is often separated from state affairs, and scholars who vocally correct rulers are frequently marginalized. In this context, we must be vigilant in identifying true scholars:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: left\"\u003eAspect\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003eScholars: Heirs of the Prophets (Rabbani)\u003c/th\u003e\n \u003cth style=\"text-align: left\"\u003eScholars: Heirs of Desires (Su\u0026rsquo;)\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003e\u003cstrong\u003eStance on Injustice\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eVocally proclaims the truth (\u003cem\u003eal-haq\u003c/em\u003e) and stands at the forefront to prevent tyranny.\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eChooses silence, seeks justifications for injustice, or serves as a rubber stamp for tyrannical policies.\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003e\u003cstrong\u003eRelation to Rulers\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eMaintains spiritual distance (\u003cem\u003ewara\u0026rsquo;\u003c/em\u003e), counsels sincerely, and rejects bribes of offices or funding.\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eFlatters power to obtain worldly benefits, projects, positions, or financial aids.\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: left\"\u003e\u003cstrong\u003eSource of Authority\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eRecognized by the depth of their knowledge, piety, and consistency in defending the community.\u003c/td\u003e\n \u003ctd style=\"text-align: left\"\u003eDetermined solely by organizational positions or media-engineered popularity.\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-danger-of-following-devout-but-insincere-scholars\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-danger-of-following-devout-but-insincere-scholars\" class=\"header-mark\" aria-label=\"Direct link to section: The Danger of Following Devout but Insincere Scholars\"\u003e\u003c/a\u003eThe Danger of Following Devout but Insincere Scholars\u003c/h2\u003e\u003cp\u003eFollowing scholars who serve their own desires only leads the community into misguidance. In the Hereafter, blind fanaticism to such role models cannot be used as an excuse to escape punishment. Allah SWT depicts the regrets of the followers on the Day of Resurrection:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِذْ تَبَرَّأَ الَّذِينَ اتُّبِعُوا مِنَ الَّذِينَ اتَّبَعُوا وَرَأَوُا الْعَذَابَ وَتَقَطَّعَتْ بِهِمُ الْأَسْبَابُ\u0026hellip;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026quot;[And they should consider that] when those who have been followed disassociate themselves from those who followed them, and they [all] see the punishment, and cut off from them are the ties [of relationship]. Those who followed will say, \u0026lsquo;If only we had another turn [at worldly life] so we could disassociate ourselves from them as they have disassociated themselves from us.\u0026rsquo;\u0026hellip;\u0026quot;\u003c/em\u003e (QS. Al-Baqarah: 166-167)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eTrue scholars are the lanterns of the community who bring tranquility and truth. They cannot be bought with material gains nor silenced with threats. May Allah SWT always open our hearts to distinguish between scholars who inherit the prophetic mission and those who inherit worldly desires, and protect the rabbani scholars from all evil schemes.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Sodiq Fajar (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "The Qur'an and Sunnah: Guideline and Soul of Life", "date_published" : "2026-07-16T21:02:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/al-quran-dan-as-sunnah-pedoman-dan-ruh-kehidupan/", "url" : "https://www.aantriono.com/en/al-quran-dan-as-sunnah-pedoman-dan-ruh-kehidupan/", "summary": " IntroductionAbsolutely, every human being requires a firm grip and a guide when navigating life in this world. Without a clear guide, one will be tossed around like a feather in a strong wind—directionless, confused, and unsure of their final destination. For a Muslim, the ultimate goal of this life is already very clear: to attain happiness in this world and salvation in the Hereafter, seeking the pleasure of Allah SWT.\nTo achieve this noble goal, Allah SWT does not leave mankind to wander in the dark. He obliges Muslims to make the Qur\u0026rsquo;an and Sunnah their primary guide. More than just a guidebook, both must penetrate to become the soul of life that drives our every breath and daily activities.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eAbsolutely, every human being requires a firm grip and a guide when navigating life in this world. Without a clear guide, one will be tossed around like a feather in a strong wind—directionless, confused, and unsure of their final destination. For a Muslim, the ultimate goal of this life is already very clear: to attain happiness in this world and salvation in the Hereafter, seeking the pleasure of Allah SWT.\u003c/p\u003e\n\u003cp\u003eTo achieve this noble goal, Allah SWT does not leave mankind to wander in the dark. He obliges Muslims to make the Qur\u0026rsquo;an and Sunnah their primary guide. More than just a guidebook, both must penetrate to become the \u003cstrong\u003esoul of life\u003c/strong\u003e that drives our every breath and daily activities.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-quran-as-soul-and-light\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-quran-as-soul-and-light\" class=\"header-mark\" aria-label=\"Direct link to section: The Qur\u0026rsquo;an as Soul and Light\"\u003e\u003c/a\u003eThe Qur\u0026rsquo;an as Soul and Light\u003c/h2\u003e\u003cp\u003eIn the Qur\u0026rsquo;an, Allah SWT terms His revelation as \u0026ldquo;ruh\u0026rdquo; (soul/spirit) because it provides spiritual life to human souls that were previously dead in the darkness of ignorance (Jahiliyah):\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَكَذٰلِكَ اَوْحَيْنَآ اِلَيْكَ رُوْحًا مِّنْ اَمْرِنَا ۗمَا كُنْتَ تَدْرِيْ مَا الْكِتٰبُ وَلَا الْاِيْمَانُ وَلٰكِنْ جَعَلْنٰهُ نُوْرًا نَّهْدِيْ بِهٖ مَنْ نَّشَاۤءُ مِنْ عِبَادِنَا ۗوَاِنَّكَ لَتَهْدِيْٓ اِلٰى صِرَاطٍ مُّسْتَقِيْمٍۙ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And thus We have revealed to you an inspiration (ruh) of Our command. You did not know what is the Book or [what is] faith, but We have made it a light by which We guide whom We will of Our servants. And indeed, [O Muhammad], you guide to a straight path.\u0026rdquo;\u003c/em\u003e (QS. Asy-Syura: 52)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eSo close was the connection of the Messenger of Allah\u0026rsquo;s SAW life with revelation that Ummul Mukminin Aisha radhiyallahu \u0026lsquo;anha described his character by stating: \u003cem\u003e\u0026ldquo;His character was the Qur\u0026rsquo;an.\u0026rdquo;\u003c/em\u003e He was the embodiment of the walking Qur\u0026rsquo;an on earth.\u003c/p\u003e\n\u003cp\u003eAnyone who holds fast to both is guaranteed never to go astray, as the Messenger of Allah SAW said during the Farewell Pilgrimage (Haji Wada\u0026rsquo;):\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eيَا أَيُّهَا النَّاسُ إِنِّى قَدْ تَرَكْتُ فِيكُمْ مَا إِنِ اعْتَصَمْتُمْ بِهِ فَلَنْ تَضِلُّوا أَبَدًا كِتَابَ اللَّهِ وَسُنَّةَ نَبِيِّهِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;O people, indeed I have left among you that which, if you hold fast to it, you will never go astray: the Book of Allah (Qur\u0026rsquo;an) and the Sunnah of His Prophet.\u0026rdquo;\u003c/em\u003e (HR. Al-Hakim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"three-ways-to-make-the-quran--sunnah-the-soul-of-life\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-ways-to-make-the-quran--sunnah-the-soul-of-life\" class=\"header-mark\" aria-label=\"Direct link to section: Three Ways to Make the Qur\u0026rsquo;an \u0026amp; Sunnah the Soul of Life\"\u003e\u003c/a\u003eThree Ways to Make the Qur\u0026rsquo;an \u0026amp; Sunnah the Soul of Life\u003c/h2\u003e\u003cp\u003eTo implement the Qur\u0026rsquo;an and Sunnah as the soul in our daily lives, there are three practical steps we must take:\u003c/p\u003e\n\u003ch3 id=\"1-making-them-the-primary-reference-al-marji\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-making-them-the-primary-reference-al-marji\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Making Them the Primary Reference (Al-Marji\u0026rsquo;)\"\u003e\u003c/a\u003e1. Making Them the Primary Reference (\u003cem\u003eAl-Marji\u0026rsquo;\u003c/em\u003e)\u003c/h3\u003e\u003cp\u003eWhen we face disputes, doubts, or differences of opinion in any matter, our first and final reference must be the Qur\u0026rsquo;an and Sunnah, not unrestricted human logic or mere majority opinion.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eيَاأَيُّهَا الَّذِينَ آمَنُوا أَطِيعُوا اللَّهَ وَأَطِيعُوا الرَّسُولَ وَأُولِي الْأَمْرِ مِنْكُمْ فَإِنْ تَنَازَعْتُمْ فِي شَيْءٍ فَرُدُّوهُ إِلَى اللَّهِ وَالرَّسُولِ\u0026hellip;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;O you who have believed, obey Allah and obey the Messenger and those in authority among you. And if you disagree over anything, refer it to Allah (Qur\u0026rsquo;an) and the Messenger (Sunnah)\u0026hellip;\u0026rdquo;\u003c/em\u003e (QS. An-Nisa’: 59)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"2-making-them-the-standard-of-judgment-al-miyar\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-making-them-the-standard-of-judgment-al-miyar\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Making Them the Standard of Judgment (Al-Miyar)\"\u003e\u003c/a\u003e2. Making Them the Standard of Judgment (\u003cem\u003eAl-Miyar\u003c/em\u003e)\u003c/h3\u003e\u003cp\u003eThe Qur\u0026rsquo;an and Sunnah must be positioned as the yardstick (standard) to judge modern thoughts, cultures, customs, philosophies, and laws. Modern culture or rules must submit and adapt to Islam, not the other way around where Islamic teachings are altered or cropped to suit the tastes of the times.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَمَا اخْتَلَفْتُمْ فِيهِ مِنْ شَيْءٍ فَحُكْمُهُ إِلَى اللَّهِ\u0026hellip;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And in anything over which you disagree—its ruling is [to be referred] to Allah\u0026hellip;\u0026rdquo;\u003c/em\u003e (QS. Asy-Syura: 10)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"3-applying-their-laws-collectively-and-comprehensively\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-applying-their-laws-collectively-and-comprehensively\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Applying Their Laws Collectively and Comprehensively\"\u003e\u003c/a\u003e3. Applying Their Laws Collectively and Comprehensively\u003c/h3\u003e\u003cp\u003eWe must not be selective (partial) towards Islamic teachings—accepting rules that suit our desires and discarding those deemed heavy or difficult. We are obliged to submit to all His laws in all aspects of life, including worship, morality, social transactions, and criminal laws.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"learning-from-the-best-generation-the-sahabah\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-from-the-best-generation-the-sahabah\" class=\"header-mark\" aria-label=\"Direct link to section: Learning from the Best Generation (The Sahabah)\"\u003e\u003c/a\u003eLearning from the Best Generation (The Sahabah)\u003c/h2\u003e\u003cp\u003eIn his book of exegesis \u003cem\u003eFi Zhilalil Qur\u0026rsquo;an\u003c/em\u003e, Sayyid Qutb explained three main secrets why the generation of the companions radhiyallahu \u0026lsquo;anhum succeeded in becoming the best generation, unmatched in human history:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003ePurity of Source:\u003c/strong\u003e They made the Qur\u0026rsquo;an their sole source of guidance and discarded all other foreign sources of thought.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAction-Oriented:\u003c/strong\u003e They read the Qur\u0026rsquo;an not merely for academic knowledge (\u003cem\u003etsaqafah\u003c/em\u003e) or literary appreciation, but directly to implement it in real life.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBreak from the Past:\u003c/strong\u003e They completely severed all emotional and intellectual ties to their past ignorance (Jahiliyah) and began an entirely new chapter of life with Islam.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe honor of the Muslim Ummah depends heavily on how firmly they hold fast to this magnificent guide. The Prophet SAW warned in his hadith:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِنَّ اللَّهَ يَرْفَعُ بِهَذَا الْكِتَابِ أَقْوَامًا وَيَضَعُ بِهِ آخَرِيْنَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Indeed, Allah elevates nations with this Book (the Qur\u0026rsquo;an) and degrades others with it.\u0026rdquo;\u003c/em\u003e (HR. Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eLet us make the Qur\u0026rsquo;an and Sunnah the guiding light of our footsteps and the soul that revives our hearts.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Hamid S. (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Charity of Water During the Dry Season", "date_published" : "2026-07-16T20:57:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sedekah-air-di-musim-kemarau/", "url" : "https://www.aantriono.com/en/sedekah-air-di-musim-kemarau/", "summary": " IntroductionA prolonged dry season often brings severe trials for many of our brothers and sisters. The scorching sun, dry and dusty air, dried-up fields, and dried-up household wells cut off clean water supplies. Household chores are disrupted, and the livelihood of many farmers is halted due to crop failure. It is in times like these that we realize how precious the blessing of water is—a blessing we often take for granted.\nIn the Qur\u0026rsquo;an, Allah SWT repeatedly emphasizes that water is the primary source of life:\nوَجَعَلْنَا مِنَ الْمَاۤءِ كُلَّ شَيْءٍ حَيٍّۗ اَفَلَا يُؤْمِنُوْنَ\nMeaning: \u0026quot;\u0026hellip;and We made from water every living thing? Then will they not believe?\u0026quot; (QS. Al-Anbiyā`: 30)\nAllah also invites humanity to reflect on the blessing of clean, drinkable water:\nاَفَرَءَيْتُمُ الْمَاۤءَ الَّذِيْ تَشْرَبُوْنَۗ ءَاَنْتُمْ اَنْزَلْتُمُوْهُ مِنَ الْمُزْنِ اَمْ نَحْنُ الْمُنْزِلُوْنَ لَوْ نَشَاۤءُ جَعَلْنٰهُ اُجَاجًا فَلَوْلَا تَشْكُرُوْنَ\nMeaning: \u0026ldquo;Have you seen the water that you drink? Is it you who brought it down from the clouds, or is it We who bring it down? If We willed, We could make it salty (and bitter), so why are you not grateful?\u0026rdquo; (QS. Al-Wāqi’ah: 68-70)\nOne of the best ways to express gratitude for the abundance of water we enjoy is by sharing it with those in need through water charity.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eA prolonged dry season often brings severe trials for many of our brothers and sisters. The scorching sun, dry and dusty air, dried-up fields, and dried-up household wells cut off clean water supplies. Household chores are disrupted, and the livelihood of many farmers is halted due to crop failure. It is in times like these that we realize how precious the blessing of water is—a blessing we often take for granted.\u003c/p\u003e\n\u003cp\u003eIn the Qur\u0026rsquo;an, Allah SWT repeatedly emphasizes that water is the primary source of life:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَجَعَلْنَا مِنَ الْمَاۤءِ كُلَّ شَيْءٍ حَيٍّۗ اَفَلَا يُؤْمِنُوْنَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026quot;\u0026hellip;and We made from water every living thing? Then will they not believe?\u0026quot;\u003c/em\u003e (QS. Al-Anbiyā`: 30)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eAllah also invites humanity to reflect on the blessing of clean, drinkable water:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eاَفَرَءَيْتُمُ الْمَاۤءَ الَّذِيْ تَشْرَبُوْنَۗ ءَاَنْتُمْ اَنْزَلْتُمُوْهُ مِنَ الْمُزْنِ اَمْ نَحْنُ الْمُنْزِلُوْنَ لَوْ نَشَاۤءُ جَعَلْنٰهُ اُجَاجًا فَلَوْلَا تَشْكُرُوْنَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Have you seen the water that you drink? Is it you who brought it down from the clouds, or is it We who bring it down? If We willed, We could make it salty (and bitter), so why are you not grateful?\u0026rdquo;\u003c/em\u003e (QS. Al-Wāqi’ah: 68-70)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eOne of the best ways to express gratitude for the abundance of water we enjoy is by sharing it with those in need through \u003cstrong\u003ewater charity\u003c/strong\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"three-extraordinary-virtues-of-water-charity\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-extraordinary-virtues-of-water-charity\" class=\"header-mark\" aria-label=\"Direct link to section: Three Extraordinary Virtues of Water Charity\"\u003e\u003c/a\u003eThree Extraordinary Virtues of Water Charity\u003c/h2\u003e\u003cp\u003eProviding water for those who are thirsty or for regions struck by drought holds immense virtues in the sight of Allah SWT and His Messenger.\u003c/p\u003e\n\u003ch3 id=\"1-the-most-virtuous-charity\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-the-most-virtuous-charity\" class=\"header-mark\" aria-label=\"Direct link to section: 1. The Most Virtuous Charity\"\u003e\u003c/a\u003e1. The Most Virtuous Charity\u003c/h3\u003e\u003cp\u003eWhen the mother of the companion Sa\u0026rsquo;d bin Ubadah radhiyallahu \u0026lsquo;anhu passed away, he wished to dedicate a continuous charity (\u003cem\u003esadaqah jariyah\u003c/em\u003e) on her behalf. He asked the Prophet SAW:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eيَا رَسُولَ اللَّهِ، إِنَّ أُمِّي مَاتَتْ أَفَأَتَصَدَّقُ عَنْهَا؟ قَالَ: نَعَمْ، قُلْتُ: فَأَيُّ الصَّدَقَةِ أَفْضَلُ؟ قَالَ: سَقْيُ الْمَاءِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;O Messenger of Allah, my mother has passed away. May I give charity on her behalf?\u0026rdquo; The Prophet replied, \u0026ldquo;Yes.\u0026rdquo; Sa\u0026rsquo;d asked again, \u0026ldquo;Which charity is the most virtuous?\u0026rdquo; The Prophet answered, \u0026ldquo;Providing water.\u0026rdquo;\u003c/em\u003e (HR. An-Nasai No. 3664)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"2-a-special-drink-in-paradise\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-a-special-drink-in-paradise\" class=\"header-mark\" aria-label=\"Direct link to section: 2. A Special Drink in Paradise\"\u003e\u003c/a\u003e2. A Special Drink in Paradise\u003c/h3\u003e\u003cp\u003eEvery drop of water we provide to ease someone\u0026rsquo;s thirst in this world will be rewarded by Allah with the best drink in the Hereafter:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eأَيَّمَا مُسْلِمٍ سَقَى مُسْلِمًا عَلَى ظَمَأ سَقَاهُ اللَّهُ مِنَ الرَّحِيقِ الْمَخْتُومِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Whichever Muslim gives a drink to another Muslim who is thirsty, Allah will give him to drink from Ar-Rahiq Al-Makhtum (sealed pure drink in Paradise).\u0026rdquo;\u003c/em\u003e (HR. Abu Daud No. 1682)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"3-a-means-of-forgiveness-and-entry-to-paradise\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-a-means-of-forgiveness-and-entry-to-paradise\" class=\"header-mark\" aria-label=\"Direct link to section: 3. A Means of Forgiveness and Entry to Paradise\"\u003e\u003c/a\u003e3. A Means of Forgiveness and Entry to Paradise\u003c/h3\u003e\u003cp\u003eCharity of water, even to creatures other than humans, carries weight on the Scales. We have often heard the authentic narration about a man who took pity on a thirsty dog and gave it water using his leather sock. The Prophet SAW stated that because of his action:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eفَشَكَرَ اللَّهُ لَهُ فَغَفَرَ لَهُ. قَالُوا: يَا رَسُولَ اللَّهِ وَإِنَّ لَنَا فِي الْبَهَائِمِ أَجْرًا؟ قَالَ: فِي كُلِّ كَبِدٍ رَطْبَةٍ أَجْرٌ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026quot;\u0026hellip;so Allah thanked him and forgave him.\u0026quot; The companions asked, \u0026ldquo;O Messenger of Allah, do we get a reward for serving animals?\u0026rdquo; He replied, \u0026ldquo;There is a reward for serving any living being (with a moist liver).\u0026rdquo;\u003c/em\u003e (HR. Al-Bukhari \u0026amp; Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"concrete-forms-of-water-charity\" class=\"headerLink\"\u003e\n \u003ca href=\"#concrete-forms-of-water-charity\" class=\"header-mark\" aria-label=\"Direct link to section: Concrete Forms of Water Charity\"\u003e\u003c/a\u003eConcrete Forms of Water Charity\u003c/h2\u003e\u003cp\u003eWater charity is not limited to distributing bottled water or emergency water trucks to drought-hit areas. The benefit multiplies when it is realized in the form of long-term projects, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eDrilling Wells:\u003c/strong\u003e Constructing deep borewells in dry communities so water remains continuously available for daily needs.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBuilding Reservoirs/Water Pools:\u003c/strong\u003e Similar to Sa\u0026rsquo;d bin Ubadah who built pools to collect water for the people of Medina.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"water-charity-as-a-means-of-healing\" class=\"headerLink\"\u003e\n \u003ca href=\"#water-charity-as-a-means-of-healing\" class=\"header-mark\" aria-label=\"Direct link to section: Water Charity as a Means of Healing\"\u003e\u003c/a\u003eWater Charity as a Means of Healing\u003c/h3\u003e\u003cp\u003eIn \u003cem\u003eSiyar A\u0026rsquo;lam an-Nubala\u0026rsquo;\u003c/em\u003e (8/408), Imam adz-Dzahabi recorded that a man complained of a knee wound that had not healed for 7 years to the scholar Ibnul Mubarak. After various medical treatments failed, Ibnul Mubarak advised him:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;Go and dig a well in a place where people are in desperate need of water. I hope that water will gush forth there and stop the bleeding of your wound.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe man did so, and by Allah\u0026rsquo;s leave, his knee wound healed completely after the well was dug.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eCharity of water during the dry season may seem simple to us, but in the sight of Allah SWT, this deed is of grand status. It is a tangible manifestation of social solidarity (\u003cem\u003eukhuwah\u003c/em\u003e) and a proof of our piety. In this dry season, let us unite to alleviate the burden of our brothers and sisters who lack water, so that Allah SWT may also quench our thirst on the Day of Resurrection.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Sodiq Fajar (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Islam Arrives to Honor Humanity", "date_published" : "2026-07-16T20:52:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/islam-hadir-memuliakan-manusia/", "url" : "https://www.aantriono.com/en/islam-hadir-memuliakan-manusia/", "summary": " IntroductionThe colorful life of this world often distracts human beings. Amidst the busyness of hoarding wealth, pursuing careers, and fulfilling material desires, we often forget the fundamental essence of our existence on this earth. Allah SWT reminds us of the purpose of human creation in the Qur\u0026rsquo;an:\nوَمَا خَلَقْتُ الْجِنَّ وَالْاِنْسَ اِلَّا لِيَعْبُدُوْنِ\nMeaning: \u0026ldquo;And I did not create the jinn and mankind except to worship Me.\u0026rdquo; (QS. Adz-Dzariyat: 56)\nThe world we live in today is very different from Paradise—the original home of our ancestor, Adam \u0026lsquo;alaihissalam. Paradise is a place of eternal delight where all desires are permitted without any restriction. On the contrary, the world is designed as a temporary testing ground, filled with regulations, boundaries, and hard work.\nIn order for this dynamic worldly life to run harmoniously and to secure our safety in the Hereafter, Allah SWT sent down the Islamic Sharia. Islam is here not to cause hardship, but to honor humanity.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThe colorful life of this world often distracts human beings. Amidst the busyness of hoarding wealth, pursuing careers, and fulfilling material desires, we often forget the fundamental essence of our existence on this earth. Allah SWT reminds us of the purpose of human creation in the Qur\u0026rsquo;an:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَمَا خَلَقْتُ الْجِنَّ وَالْاِنْسَ اِلَّا لِيَعْبُدُوْنِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And I did not create the jinn and mankind except to worship Me.\u0026rdquo;\u003c/em\u003e (QS. Adz-Dzariyat: 56)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe world we live in today is very different from Paradise—the original home of our ancestor, Adam \u0026lsquo;alaihissalam. Paradise is a place of eternal delight where all desires are permitted without any restriction. On the contrary, the world is designed as a temporary testing ground, filled with regulations, boundaries, and hard work.\u003c/p\u003e\n\u003cp\u003eIn order for this dynamic worldly life to run harmoniously and to secure our safety in the Hereafter, Allah SWT sent down the Islamic Sharia. Islam is here not to cause hardship, but to honor humanity.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"allahs-guidance-dispeller-of-fear-and-grief\" class=\"headerLink\"\u003e\n \u003ca href=\"#allahs-guidance-dispeller-of-fear-and-grief\" class=\"header-mark\" aria-label=\"Direct link to section: Allah\u0026rsquo;s Guidance: Dispeller of Fear and Grief\"\u003e\u003c/a\u003eAllah\u0026rsquo;s Guidance: Dispeller of Fear and Grief\u003c/h2\u003e\u003cp\u003eWhen Adam was sent down to earth, Allah promised guidance for his descendants so they would be safe in navigating worldly life.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eقُلْنَا اهْبِطُوْا مِنْهَا جَمِيْعًا ۚ فَاِمَّا يَأْتِيَنَّكُمْ مِّنِّيْ هُدًى فَمَنْ تَبِعَ هُدَايَ فَلَا خَوْفٌ عَلَيْهِمْ وَلَا هُمْ يَحْزَنُوْنَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;We said, \u0026lsquo;Go down from it, all of you. And when guidance comes to you from Me, whoever follows My guidance—there will be no fear concerning them, nor will they grieve.\u0026rsquo;\u0026rdquo;\u003c/em\u003e (QS. Al-Baqarah: 38)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eObedience to Allah\u0026rsquo;s Sharia brings immense peace of mind in this world and happiness in the Hereafter. Believers do not suffer from excessive anxiety because they entrust all their affairs to the All-Rich, All-Wise Lord.\u003c/p\u003e\n\u003cp\u003eConversely, ignoring Allah\u0026rsquo;s Sharia and indulging in sins only breeds fear, anxiety, and distress in the human heart:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eسَنُلْقِيْ فِيْ قُلُوْبِ الَّذِيْنَ كَفَرُوا الرُّعْبَ بِمَٓا اَشْرَكُوْا بِاللّٰهِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;We will cast terror into the hearts of those who disbelieve because they have associated with Allah\u0026hellip;\u0026rdquo;\u003c/em\u003e (QS. Ali Imran: 151)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"every-command-is-a-blessing-every-prohibition-is-a-shield\" class=\"headerLink\"\u003e\n \u003ca href=\"#every-command-is-a-blessing-every-prohibition-is-a-shield\" class=\"header-mark\" aria-label=\"Direct link to section: Every Command Is a Blessing, Every Prohibition Is a Shield\"\u003e\u003c/a\u003eEvery Command Is a Blessing, Every Prohibition Is a Shield\u003c/h2\u003e\u003cp\u003eWhen we reflect on every rule in Islam, we realize that all Sharia laws aim for the benefit of humanity itself.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eAllah\u0026rsquo;s Commands\u003c/strong\u003e are revealed to cleanse the soul and perfect the blessings of life:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eمَا يُرِيْدُ اللّٰهُ لِيَجْعَلَ عَلَيْكُمْ مِّنْ حَرَجٍ وَّلٰكِنْ يُّرِيْدُ لِيُطَهِّرَكُمْ وَلِيُتِمَّ نِعْمَتَهٗ عَلَيْكُمْ لَعَلَّكُمْ تَشْكُرُوْنَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Allah does not intend to make difficulty for you, but He intends to purify you and complete His favor upon you that you may be grateful.\u0026rdquo;\u003c/em\u003e (QS. Al-Maidah: 6)\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eAllah\u0026rsquo;s Prohibitions\u003c/strong\u003e are established to protect humans from dangers that are far greater than any temporary benefit. For example, the prohibition of alcohol (khamr) and gambling:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eيَسْـَٔلُوْنَكَ عَنِ الْخَمْرِ وَالْمَيْسِرِۗ قُلْ فِيْهِمَآ اِثْمٌ كَبِيْرٌ وَّمَنَافِعُ لِلنَّاسِۖ وَاِثْمُهُمَآ اَكْبَرُ مِنْ نَّفْعِهِمَاۗ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;They ask you about wine and gambling. Say, \u0026lsquo;In them is great sin and [yet, some] benefit for people. But their sin is greater than their benefit.\u0026rsquo;\u0026rdquo;\u003c/em\u003e (QS. Al-Baqarah: 219)\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"patience-the-key-to-true-honor\" class=\"headerLink\"\u003e\n \u003ca href=\"#patience-the-key-to-true-honor\" class=\"header-mark\" aria-label=\"Direct link to section: Patience: The Key to True Honor\"\u003e\u003c/a\u003ePatience: The Key to True Honor\u003c/h2\u003e\u003cp\u003eThe ultimate key for humanity to achieve honor in this world and the Hereafter is patience in obeying Allah and always maintaining a good opinion (\u003cem\u003ehusnuzhan\u003c/em\u003e) of His decrees. When life\u0026rsquo;s trials strike, know that Allah has prepared ease to follow:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eفَاِنَّ مَعَ الْعُسْرِ يُسْرًاۙ اِنَّ مَعَ الْعُسْرِ يُسْرًاۗ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;For indeed, with hardship [will be] ease. Indeed, with hardship [will be] ease.\u0026rdquo;\u003c/em\u003e (QS. Al-Insyirah: 5-6)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eHaving a bad opinion of Allah when tested will only plunge a person into clear ruin in both worlds. Allah SWT says:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَمِنَ النَّاسِ مَنْ يَّعْبُدُ اللّٰهَ عَلٰى حَرْفٍۚ فَاِنْ اَصَابَهٗ خَيْرُ ِۨاطْمَئَنَّ بِهٖۚ وَاِنْ اَصَابَتْهُ فِتْنَةُ ِۨانْقَلَبَ عَلٰى وَجْهِهٖۗ خَسِرَ الدُّنْيَا وَالْاٰخِرَةَۗ ذٰلِكَ هُوَ الْخُسْرَانُ الْمُبِيْنُ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And of the people is he who worships Allah on an edge. If he is touched by good, he is reassured by it; but if he is struck by trial, he turns on his face. He has lost [both] this world and the Hereafter. That is the clear loss.\u0026rdquo;\u003c/em\u003e (QS. Al-Hajj: 11)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eIslam is not a shackle that limits human freedom, but a compass that guides humanity to live an honorable and dignified life. By complying with His commands and avoiding His prohibitions, we are preserving the purity of our souls and securing eternal happiness in the Hereafter. May Allah grant us steadfastness to always stay under the shade of His magnificent guidance.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Fahrurozi, M.Pd. (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Three Most Powerful Dhikr Recitations", "date_published" : "2026-07-16T20:48:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/tiga-bacaan-zikir-paling-dahsyat/", "url" : "https://www.aantriono.com/en/tiga-bacaan-zikir-paling-dahsyat/", "summary": " IntroductionRemembering Allah (dhikrullah) is one of the most noble, easy, yet spiritually powerful tongue-based acts of worship. Allah SWT commands His servants to remember Him abundantly, reminding us that through the remembrance of Allah, the human heart finds ultimate peace and tranquility.\nAmong the various dhikr taught by the Prophet Muhammad SAW, there are three primary recitations recommended to be practiced 100 times daily. These dhikr possess immense power, offering grand-scale forgiveness of sins and the multiplication of good deeds.\nHere are the three most powerful dhikr recitations along with their respective virtues:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eRemembering Allah (\u003cem\u003edhikrullah\u003c/em\u003e) is one of the most noble, easy, yet spiritually powerful tongue-based acts of worship. Allah SWT commands His servants to remember Him abundantly, reminding us that through the remembrance of Allah, the human heart finds ultimate peace and tranquility.\u003c/p\u003e\n\u003cp\u003eAmong the various dhikr taught by the Prophet Muhammad SAW, there are three primary recitations recommended to be practiced \u003cstrong\u003e100 times daily\u003c/strong\u003e. These dhikr possess immense power, offering grand-scale forgiveness of sins and the multiplication of good deeds.\u003c/p\u003e\n\u003cp\u003eHere are the three most powerful dhikr recitations along with their respective virtues:\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"1-the-first-dhikr-full-tahlil\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-the-first-dhikr-full-tahlil\" class=\"header-mark\" aria-label=\"Direct link to section: 1. The First Dhikr: Full Tahlil\"\u003e\u003c/a\u003e1. The First Dhikr: Full Tahlil\u003c/h2\u003e\u003cp\u003eThe tahlil formulation to recite is:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eلَا إِلَهَ إِلَّا اللهُ وَحْدَهُ لَا شَرِيكَ لَهُ لَهُ الْمُلْكُ وَلَهُ الْحَمْدُ وَهُوَ عَلَى ‌كُلِّ ‌شَيْءٍ ‌قَدِيرٌ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;There is no deity worthy of worship except Allah, He is Alone, without partner. To Him belongs all sovereignty and praise, and He has power over all things.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"virtues\" class=\"headerLink\"\u003e\n \u003ca href=\"#virtues\" class=\"header-mark\" aria-label=\"Direct link to section: Virtues:\"\u003e\u003c/a\u003eVirtues:\u003c/h3\u003e\u003cp\u003eThe Prophet SAW said in a narration recorded by Imam Al-Bukhari (No. 3293):\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;Whoever says this \u003cstrong\u003eone hundred times\u003c/strong\u003e in a day will have a reward equivalent to \u003cstrong\u003efreeing ten slaves\u003c/strong\u003e, \u003cstrong\u003eone hundred good deeds\u003c/strong\u003e will be recorded for him, \u003cstrong\u003eone hundred bad deeds\u003c/strong\u003e will be erased from his record, and he will have \u003cstrong\u003eprotection from Satan\u003c/strong\u003e on that day until evening. No one will bring a better deed than him except someone who does more than that.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"2-the-second-dhikr-tasbih-subhanallah-wa-bihamdihi\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-the-second-dhikr-tasbih-subhanallah-wa-bihamdihi\" class=\"header-mark\" aria-label=\"Direct link to section: 2. The Second Dhikr: Tasbih (Subhanallah wa Bihamdihi)\"\u003e\u003c/a\u003e2. The Second Dhikr: Tasbih (Subhanallah wa Bihamdihi)\u003c/h2\u003e\u003cp\u003eThe tasbih formulation to recite is:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e‌سُبْحَانَ ‌اللهِ ‌وَبِحَمْدِهِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Glory be to Allah and all praise is to Him.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"virtues-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#virtues-1\" class=\"header-mark\" aria-label=\"Direct link to section: Virtues:\"\u003e\u003c/a\u003eVirtues:\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eForgiveness Like the Foam of the Sea:\u003c/strong\u003e The Prophet SAW said in a hadith narrated by Al-Bukhari (No. 6405):\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;Whoever recites \u0026lsquo;Subhanallah wa bihamdihi\u0026rsquo; \u003cstrong\u003eone hundred times\u003c/strong\u003e a day will have his sins forgiven, even if they are like the foam of the sea.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eGaining a Thousand Good Deeds a Day:\u003c/strong\u003e One day, the Prophet SAW asked his companions, \u003cem\u003e\u0026ldquo;Is any of you unable to earn a thousand good deeds in a day?\u0026rdquo;\u003c/em\u003e One of them asked, \u003cem\u003e\u0026ldquo;How can we earn a thousand good deeds, O Messenger of Allah?\u0026rdquo;\u003c/em\u003e He replied:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;By glorifying Allah (saying Subhanallah) \u003cstrong\u003eone hundred times\u003c/strong\u003e, for which \u003cstrong\u003ea thousand good deeds\u003c/strong\u003e will be written for him or \u003cstrong\u003ea thousand sins\u003c/strong\u003e will be wiped off from his record.\u0026rdquo;\u003c/em\u003e (HR. Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003chr\u003e\n\u003ch2 id=\"3-the-third-dhikr-istighfar-and-repentance\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-the-third-dhikr-istighfar-and-repentance\" class=\"header-mark\" aria-label=\"Direct link to section: 3. The Third Dhikr: Istighfar and Repentance\"\u003e\u003c/a\u003e3. The Third Dhikr: Istighfar and Repentance\u003c/h2\u003e\u003cp\u003eThe formulation of istighfar frequently recited by the Prophet SAW is:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e‌رَبِّ ‌اغْفِرْ ‌لِي ‌وَتُب ‌عَلَيَّ ‌إِنَّكَ ‌أَنْتَ ‌التَّوَّابُ ‌الرَّحِيمُ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;My Lord, forgive me and accept my repentance. Verily, You are the One Who accepts repentance, the Merciful.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch3 id=\"virtues-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#virtues-2\" class=\"header-mark\" aria-label=\"Direct link to section: Virtues:\"\u003e\u003c/a\u003eVirtues:\u003c/h3\u003e\u003cp\u003eEven though the Prophet SAW was infallible (\u003cem\u003ema\u0026rsquo;sum\u003c/em\u003e), he was the most diligent in seeking forgiveness. The companion Ibn Umar radhiyallahu \u0026lsquo;anhuma related:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;We counted that the Messenger of Allah shallallahu \u0026lsquo;alaihi wasallam would say in a single gathering: \u0026lsquo;Rabbighfirlii wa tub \u0026lsquo;alayya innaka antat tawwabur rahiim\u0026rsquo; \u003cstrong\u003eone hundred times\u003c/strong\u003e.\u0026rdquo;\u003c/em\u003e (HR. Ibnu Majah No. 3814)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe Prophet SAW also said:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cem\u003e\u0026ldquo;O people, repent to Allah! Verily, I repent to Him \u003cstrong\u003eone hundred times\u003c/strong\u003e in a day.\u0026rdquo;\u003c/em\u003e (HR. Muslim No. 2702)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThese three dhikr are extremely light on the tongue but weigh heavily in the Scales of Allah SWT. Allocating just 10 to 15 minutes daily to complete the count of one hundred for each of these recitations is one of the best investments for the Hereafter. Let us moisten our tongues daily with tahlil, tasbih, and istighfar, as a testament to our love for Allah SWT and His Messenger.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Sodiq Fajar (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Beware of Thoughts That Destroy Faith", "date_published" : "2026-07-16T20:45:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/waspada-pemikiran-perusak-iman/", "url" : "https://www.aantriono.com/en/waspada-pemikiran-perusak-iman/", "summary": " IntroductionIslam is a religion that highly respects the intellect and knowledge. The ability to think is one of the greatest distinctions of human beings, separating them from other creations. Therefore, the Qur\u0026rsquo;an consistently encourages humanity to use their intellect to understand the signs of Allah\u0026rsquo;s greatness, ponder His creation, and extract wisdom from every event in life.\nIn the language of the Qur\u0026rsquo;an, those who use their minds well are referred to as Ulul Albab (people of understanding) or Ulin Nuha (people of intelligence). Conversely, those who refuse to think receive severe warnings and rebukes because they neglect this divine blessing. The Qur\u0026rsquo;an frequently admonishes them with the expression:\nأَفَلَا تَعْقِلُونَ\nMeaning: \u0026ldquo;Do you not use your intellect?\u0026rdquo;\nThus, it is not surprising that the very first revelation sent down to the Prophet Muhammad SAW was a command to maximize the potential of the intellect through reading, grounded in faith in the Creator:\nاِقْرَأْ بِاسْمِ رَبِّكَ الَّذِيْ خَلَقَ\nMeaning: \u0026ldquo;Recite in the name of your Lord who created.\u0026rdquo; (QS. Al-\u0026lsquo;Alaq: 1)\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIslam is a religion that highly respects the intellect and knowledge. The ability to think is one of the greatest distinctions of human beings, separating them from other creations. Therefore, the Qur\u0026rsquo;an consistently encourages humanity to use their intellect to understand the signs of Allah\u0026rsquo;s greatness, ponder His creation, and extract wisdom from every event in life.\u003c/p\u003e\n\u003cp\u003eIn the language of the Qur\u0026rsquo;an, those who use their minds well are referred to as \u003cem\u003eUlul Albab\u003c/em\u003e (people of understanding) or \u003cem\u003eUlin Nuha\u003c/em\u003e (people of intelligence). Conversely, those who refuse to think receive severe warnings and rebukes because they neglect this divine blessing. The Qur\u0026rsquo;an frequently admonishes them with the expression:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eأَفَلَا تَعْقِلُونَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Do you not use your intellect?\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThus, it is not surprising that the very first revelation sent down to the Prophet Muhammad SAW was a command to maximize the potential of the intellect through reading, grounded in faith in the Creator:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eاِقْرَأْ بِاسْمِ رَبِّكَ الَّذِيْ خَلَقَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Recite in the name of your Lord who created.\u0026rdquo;\u003c/em\u003e (QS. Al-\u0026lsquo;Alaq: 1)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"the-intellectual-challenge-root-of-the-ummahs-crisis\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-intellectual-challenge-root-of-the-ummahs-crisis\" class=\"header-mark\" aria-label=\"Direct link to section: The Intellectual Challenge: Root of the Ummah\u0026rsquo;s Crisis\"\u003e\u003c/a\u003eThe Intellectual Challenge: Root of the Ummah\u0026rsquo;s Crisis\u003c/h2\u003e\u003cp\u003eGiven the vital role of the intellect, the most fundamental challenge facing the Muslim Ummah today is not merely economic, political, social, or military, but rather \u003cstrong\u003eintellectual challenges (intellectual warfare / \u003cem\u003eghazwul fikri\u003c/em\u003e)\u003c/strong\u003e. Various problems arising in political and social spheres, when traced to their roots, often originate from intellectual confusion.\u003c/p\u003e\n\u003cp\u003eThis intellectual challenge comes from two directions: internal and external.\u003c/p\u003e\n\u003ch3 id=\"1-internal-challenges\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-internal-challenges\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Internal Challenges\"\u003e\u003c/a\u003e1. Internal Challenges\u003c/h3\u003e\u003cp\u003eChallenges from within the Muslim Ummah include \u003cem\u003ekejumudan\u003c/em\u003e (intellectual stagnation), blind fanaticism, \u003cem\u003etaqlid\u003c/em\u003e (blind imitation without proof), and the spread of innovations (\u003cem\u003ebid\u0026rsquo;ah\u003c/em\u003e) and myths (\u003cem\u003ekhurafat\u003c/em\u003e). This condition results in careless and slow ijtihad (independent reasoning) when responding to contemporary issues.\u003c/p\u003e\n\u003ch3 id=\"2-external-challenges\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-external-challenges\" class=\"header-mark\" aria-label=\"Direct link to section: 2. External Challenges\"\u003e\u003c/a\u003e2. External Challenges\u003c/h3\u003e\u003cp\u003eChallenges from the outside involve the infiltration of foreign, secular concepts and viewpoints into Islamic religious discourse. These ideologies include \u003cstrong\u003esecularism, liberalism, religious pluralism, relativism, and feminism/gender theories\u003c/strong\u003e. As a result, cognitive distortion and intellectual confusion arise among Muslims, leading them to view Islam through a secular and relativistic Western lens.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-danger-of-ideological-invasion-in-the-ummah\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-danger-of-ideological-invasion-in-the-ummah\" class=\"header-mark\" aria-label=\"Direct link to section: The Danger of Ideological Invasion in the Ummah\"\u003e\u003c/a\u003eThe Danger of Ideological Invasion in the Ummah\u003c/h2\u003e\u003cp\u003eThe most severe damage occurs when these destructive thoughts are adopted by some Muslim intellectuals themselves. In the name of renewal (\u003cem\u003etajdid\u003c/em\u003e), they instead propagate ideas that deviate from the fundamental principles of Sharia:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eRelativism of Truth:\u003c/strong\u003e They claim that religious values and truths are relative. Religions are deemed to have no right to claim absolute truth (\u003cem\u003etruth claim\u003c/em\u003e); instead, truth is reduced to human perceptions that shift over time.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSocial Deviation:\u003c/strong\u003e Under the guise of human rights and gender equality, some of them begin to approve same-sex marriage and challenge the Islamic inheritance laws (the 1:2 ratio between men and women), accusing it of being unjust.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSkepticism Toward Sharia Laws:\u003c/strong\u003e They view the implementation of Islamic criminal laws (\u003cem\u003ehudud\u003c/em\u003e), such as \u003cem\u003eqisas\u003c/em\u003e for murderers and stoning for married adulterers, as cruel and inhumane.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDeconstructing the Qur\u0026rsquo;an:\u003c/strong\u003e They argue that the Qur\u0026rsquo;an was revealed only to respond to the social and cultural context of ancient Arabia, and therefore its texts are no longer relevant or do not need to be practiced as they were revealed in the modern era.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThose exposed to these destructive ideologies measure everything by Western standards: viewing economics through capitalism, politics through secularism, society through modernism, and Islamic Sharia through relativism.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-stance-of-a-muslim\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-stance-of-a-muslim\" class=\"header-mark\" aria-label=\"Direct link to section: The Stance of a Muslim\"\u003e\u003c/a\u003eThe Stance of a Muslim\u003c/h2\u003e\u003cp\u003eThis intellectual challenge is a real test of faith in the modern era. History shows that the Muslim Ummah will decline if they are affected by the poison of intellectual doubts (\u003cem\u003esyubhat\u003c/em\u003e) and abandon the principles of their religion. Conversely, the Ummah will achieve glory if they always hold fast to the principles of Islamic Sharia, with their intellect guided by divine revelation.\u003c/p\u003e\n\u003cp\u003eTherefore, let us safeguard our faith by:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eContinuously Seeking Sacred Knowledge:\u003c/strong\u003e Understanding Islam from its pure sources under the guidance of trustworthy scholars to build a shield against doubts.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMaximizing a Guided Intellect:\u003c/strong\u003e Using our minds to reflect on Allah\u0026rsquo;s greatness and the perfection of His Sharia, not to challenge His absolute decrees.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBeing Wary of Foreign Ideologies:\u003c/strong\u003e Remaining critical of various modern thoughts that contradict Islamic creed and morality.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eMay Allah SWT always protect our minds and hearts from misleading thoughts that can destroy our faith, and cause us to die in a state of pure faith and Islam. \u003cem\u003eAamiin Ya Rabbal \u0026lsquo;Alamin.\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Amir Sahidin, M.Ag. (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "10 Signs of Weak Faith", "date_published" : "2026-07-16T20:37:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/10-tanda-lemahnya-iman/", "url" : "https://www.aantriono.com/en/10-tanda-lemahnya-iman/", "summary": " IntroductionIn Islamic theology, we firmly believe that faith (Iman) is volatile—it increases (yaziid) and decreases (yanqush). It grows stronger with obedience and righteous deeds, and conversely, it weakens with disobedience, negligence, and sins.\nThe weakening of faith does not happen suddenly without symptoms. There are clear signs when a person\u0026rsquo;s spiritual condition is declining. Recognizing these signs of weak faith is vital for every Muslim as a means of daily self-reflection (muhasabah) so that we can immediately correct ourselves before our faith deteriorates further.\nHere are 10 signs of weak faith that we must watch out for:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn Islamic theology, we firmly believe that faith (\u003cem\u003eIman\u003c/em\u003e) is volatile—it increases (\u003cem\u003eyaziid\u003c/em\u003e) and decreases (\u003cem\u003eyanqush\u003c/em\u003e). It grows stronger with obedience and righteous deeds, and conversely, it weakens with disobedience, negligence, and sins.\u003c/p\u003e\n\u003cp\u003eThe weakening of faith does not happen suddenly without symptoms. There are clear signs when a person\u0026rsquo;s spiritual condition is declining. Recognizing these signs of weak faith is vital for every Muslim as a means of daily self-reflection (\u003cem\u003emuhasabah\u003c/em\u003e) so that we can immediately correct ourselves before our faith deteriorates further.\u003c/p\u003e\n\u003cp\u003eHere are 10 signs of weak faith that we must watch out for:\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"1-slothfulness-in-performing-acts-of-obedience-and-underestimating-them\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-slothfulness-in-performing-acts-of-obedience-and-underestimating-them\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Slothfulness in Performing Acts of Obedience and Underestimating Them\"\u003e\u003c/a\u003e1. Slothfulness in Performing Acts of Obedience and Underestimating Them\u003c/h2\u003e\u003cp\u003eThe most visible sign of weakening faith is the emergence of extreme laziness in performing acts of worship. Obedience feels like a heavy burden, and even when performed, it is done hastily without focus or humility (\u003cem\u003ekhusyuk\u003c/em\u003e).\u003c/p\u003e\n\u003cp\u003eAllah SWT describes this spiritual laziness as one of the characteristics of the hypocrites in the Qur\u0026rsquo;an:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَإِذَا قَامُوا إِلَى الصَّلَاةِ قَامُوا كُسَالَىٰ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And when they stand for prayer, they stand lazily\u0026hellip;\u0026rdquo;\u003c/em\u003e (QS. An-Nisa’: 142)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eA person with weakening faith tends to underestimate voluntary deeds (\u003cem\u003enafilah\u003c/em\u003e), delays obligatory prayers without valid excuses, and is sluggish in attending Friday prayers. The Prophet SAW warned:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eلاَ يَزَالُ قَوْمٌ يَتَأَخَّرُونَ عَنِ الصَّفِّ الأَوَّلِ حَتَّى يُؤَّخِّرَهُمُ اللَّهُ عَزَّ وَجَلَّ فِى النَّارِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;People will continue to hold back from the first row until Allah holds them back in the Fire.\u0026rdquo;\u003c/em\u003e (HR. Abu Daud)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"2-feeling-no-weight-in-committing-disobedience-and-sins\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-feeling-no-weight-in-committing-disobedience-and-sins\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Feeling No Weight in Committing Disobedience and Sins\"\u003e\u003c/a\u003e2. Feeling No Weight in Committing Disobedience and Sins\u003c/h2\u003e\u003cp\u003eWhen faith weakens, the soul\u0026rsquo;s defense fortress crumbles. As a result, small sins are taken lightly, and disobedience is committed without guilt. If left unchecked, these sins become habits, to the point where the perpetrator dares to commit sins openly without shame.\u003c/p\u003e\n\u003cp\u003eThe Prophet SAW said:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eكُلُّ أُمَّتِى مُعَافًى إِلاَّ الْمُجَاهِرِينَ\u0026hellip;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;All of my Ummah will be forgiven except those who commit sins openly (al-mujahirin)\u0026hellip;\u0026rdquo;\u003c/em\u003e (HR. Bukhari and Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"3-not-feeling-indignant-when-sharia-rules-are-violated\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-not-feeling-indignant-when-sharia-rules-are-violated\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Not Feeling Indignant When Sharia Rules Are Violated\"\u003e\u003c/a\u003e3. Not Feeling Indignant When Sharia Rules Are Violated\u003c/h2\u003e\u003cp\u003eThe extinguishing of religious zeal (\u003cem\u003eghirah\u003c/em\u003e) in the chest is a clear sign of weak faith. A person with fragile faith remains indifferent when witnessing evils or violations of Sharia. There is no dislike in their heart, let alone an effort to change it.\u003c/p\u003e\n\u003cp\u003eThe Prophet SAW explained:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِذَا عُمِلَتْ الْخَطِيئَةُ فِي الْأَرْضِ كَانَ مَنْ شَهِدَهَا فَكَرِهَهَا وَقَالَ مَرَّةً أَنْكَرَهَا كَانَ كَمَنْ غَابَ عَنْهَا وَمَنْ غَابَ عَنْهَا فَرَضِيَهَا كَانَ كَمَنْ شَهِدَهَا\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;If a sin is committed on earth, the one who witnesses it and hates (or denounces) it is like one who was absent from it. And the one who was absent from it but approved of it is like one who witnessed it.\u0026rdquo;\u003c/em\u003e (HR. Abu Daud)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"4-only-avoiding-major-sins-while-ignoring-disliked-makruh-and-doubtful-syubhat-matters\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-only-avoiding-major-sins-while-ignoring-disliked-makruh-and-doubtful-syubhat-matters\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Only Avoiding Major Sins While Ignoring Disliked (Makruh) and Doubtful (Syubhat) Matters\"\u003e\u003c/a\u003e4. Only Avoiding Major Sins While Ignoring Disliked (Makruh) and Doubtful (Syubhat) Matters\u003c/h2\u003e\u003cp\u003eAnother sign of weak faith is when someone always looks for the minimum legal threshold. They constantly ask, \u003cem\u003e\u0026ldquo;Is this act sinful or not? Is it haram or just makruh?\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eTrivializing makruh and gray (\u003cem\u003esyubhat\u003c/em\u003e) areas will eventually drag a person into forbidden territory. The Prophet SAW likened this to a shepherd near a private pasture:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eكَالرَّاعِي يَرْعَى حَوْلَ الحِمَى يُوْشِكُ أَنْ يَرْتَعَ فِيْهِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Like a shepherd who grazes his flock around a private pasture, nearly grazing in it\u0026hellip;\u0026rdquo;\u003c/em\u003e (HR. Bukhari and Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe companion Ibn Mas\u0026rsquo;ud radhiyallahu \u0026lsquo;anhu illustrated how believers and wrongdoers view sins differently:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِنَّ الْمُؤْمِنَ يَرَى ذُنُوبَهُ كَأَنَّهُ قَاعِدٌ تَحْتَ جَبَلٍ يَخَافُ أَنْ يَقَعَ عَلَيْهِ وَإِنَّ الْفَاجِرَ يَرَى ذُنُوبَهُ كَذُبَابٍ مَرَّ عَلَى أَنْفِهِ\u0026hellip;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Indeed, the believer sees his sins as if he were sitting beneath a mountain which he fears will fall upon him. While the wicked person sees his sins as a fly that passes over his nose\u0026hellip;\u0026rdquo;\u003c/em\u003e (HR. Bukhari)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"5-feeling-chest-tightness-and-deteriorating-akhlaq-character\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-feeling-chest-tightness-and-deteriorating-akhlaq-character\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Feeling Chest Tightness and Deteriorating Akhlaq (Character)\"\u003e\u003c/a\u003e5. Feeling Chest Tightness and Deteriorating Akhlaq (Character)\u003c/h2\u003e\u003cp\u003eA decline in faith directly correlates with the loss of inner peace. The chest frequently feels tight, one becomes easily anxious over trivial matters, quick-tempered, and their character (\u003cem\u003eakhlaq\u003c/em\u003e) toward others begins to degrade. True faith, however, brings ease.\u003c/p\u003e\n\u003cp\u003eThe Prophet SAW said:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eاَلْإِيْمَانُ: اَلصَّبْرُ وَالسَّمَاحَةُ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Iman is patience and tolerance (magnanimity).\u0026rdquo;\u003c/em\u003e (Silsilah al-Ahadits ash-Shahihah)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"6-rarely-reading-the-quran-and-remembering-allah-dhikr\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-rarely-reading-the-quran-and-remembering-allah-dhikr\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Rarely Reading the Qur\u0026rsquo;an and Remembering Allah (Dhikr)\"\u003e\u003c/a\u003e6. Rarely Reading the Qur\u0026rsquo;an and Remembering Allah (Dhikr)\u003c/h2\u003e\u003cp\u003eThe Qur\u0026rsquo;an and dhikr are essential food for the soul. When faith weakens, a person loses their appetite for this spiritual nourishment. The Qur\u0026rsquo;an lies covered in dust, untouched for weeks, and the tongue becomes dry from remembering Allah. The heart hardens like a stone due to its distance from His remembrance.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"7-obsession-with-prestige-positions-and-fame\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-obsession-with-prestige-positions-and-fame\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Obsession with Prestige, Positions, and Fame\"\u003e\u003c/a\u003e7. Obsession with Prestige, Positions, and Fame\u003c/h2\u003e\u003cp\u003eThe growth of an intense craving for human praise, popularity, and authority without fear of Allah is a symptom of weak faith. A soul devoid of sincerity will always seek nourishment from the validation of creation.\u003c/p\u003e\n\u003cp\u003eThe Prophet SAW gave a stern warning about the ambition for leadership:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِنَّكُمْ سَتَحْرِصُونَ عَلَى الْإِمَارَةِ وَسَتَكُونُ نَدَامَةً يَوْمَ الْقِيَامَةِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Indeed, you will be greedy for authority, and it will be a cause of regret on the Day of Resurrection\u0026hellip;\u0026rdquo;\u003c/em\u003e (HR. Bukhari)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"8-indifference-to-the-affairs-of-muslims\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-indifference-to-the-affairs-of-muslims\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Indifference to the Affairs of Muslims\"\u003e\u003c/a\u003e8. Indifference to the Affairs of Muslims\u003c/h2\u003e\u003cp\u003eA believer with healthy faith shares the pain of their brothers and sisters. If our hearts are unaffected, indifferent, and we refuse to even pray for Muslims who are suffering from disasters or oppression around the world (such as the Uyghurs, Palestinians, or Syrians), our faith is in a critical condition.\u003c/p\u003e\n\u003cp\u003eThe Prophet SAW said:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِنَّ الْمُؤْمِنَ مِنْ أَهْلِ الْإِيمَانِ بِمَنْزِلَةِ الرَّأْسِ مِنَ الْجَسَدِ يَأْلَمُ الْمُؤْمِنُ لِأَهْلِ الْإِيمَانِ كَمَا يَأْلَمُ الْجَسَدُ لِمَا فِي الرَّأْسِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Indeed, the believer\u0026rsquo;s position among the people of faith is like the position of the head on the body; the believer feels pain for the people of faith just as the body feels pain from an ailment in the head.\u0026rdquo;\u003c/em\u003e (HR. Ahmad)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"9-lack-of-motivation-to-work-for-the-sake-of-islam\" class=\"headerLink\"\u003e\n \u003ca href=\"#9-lack-of-motivation-to-work-for-the-sake-of-islam\" class=\"header-mark\" aria-label=\"Direct link to section: 9. Lack of Motivation to Work for the Sake of Islam\"\u003e\u003c/a\u003e9. Lack of Motivation to Work for the Sake of Islam\u003c/h2\u003e\u003cp\u003eA person with weak faith tends to be spiritually and materially selfish. They are busy hoarding wealth for personal gratification but have no desire to dedicate their time, energy, intellect, or wealth to the benefit of Islamic da\u0026rsquo;wah and the community.\u003c/p\u003e\n\u003cp\u003eThis contrasts sharply with the Sahabah, such as Tsumamah bin Atsal, who immediately declared an economic embargo to defend the da\u0026rsquo;wah of the Prophet SAW right after entering Islam.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"10-excessiveness-in-worldly-matters-luxurious-living\" class=\"headerLink\"\u003e\n \u003ca href=\"#10-excessiveness-in-worldly-matters-luxurious-living\" class=\"header-mark\" aria-label=\"Direct link to section: 10. Excessiveness in Worldly Matters (Luxurious Living)\"\u003e\u003c/a\u003e10. Excessiveness in Worldly Matters (Luxurious Living)\u003c/h2\u003e\u003cp\u003eThe final sign of weak faith is drowning in a consumerist and luxurious lifestyle. One\u0026rsquo;s entire life focus is spent worrying about gourmet foods, branded clothing, expensive vehicles, and extravagant housing.\u003c/p\u003e\n\u003cp\u003eWhen sending Mu\u0026rsquo;adh bin Jabal to Yemen, the Prophet SAW advised:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِيَّاكَ وَالتَّنَعُّمَ، فَإِنَّ عِبَادَ اللهِ لَيْسُوا بِالْمُتَنَعِّمِينَ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Avoid luxury, for the servants of Allah do not live in luxury!\u0026rdquo;\u003c/em\u003e (HR. Ahmad)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eIman is the most precious asset we possess in this life. By recognizing these signs of spiritual weakness, let us perform a spiritual audit on ourselves. If we find these signs within us, let us immediately repent, tighten our worship, increase our dhikr, and ask Allah SWT to constantly renew the faith in our chests.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Sodiq Fajar (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Becoming a Grateful Servant", "date_published" : "2026-07-16T20:32:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menjadi-hamba-yang-pandai-bersyukur/", "url" : "https://www.aantriono.com/en/menjadi-hamba-yang-pandai-bersyukur/", "summary": " IntroductionGratitude is one of the greatest fruits of piety. A pious believer realizes that all the blessings they experience—whether large or seemingly simple—flow purely from the generosity of Allah SWT, the Lord of all worlds.\nFrom the moment we open our eyes in the morning, we are enveloped in an ocean of His blessings: the clean air we breathe for free, the heartbeat that continues without command, the eyes still able to behold the beauty of the world, and the tongue still granted the ability to mention His name. Yet, as humans, we often find it easier to remember a single small hardship than to ponder the thousands of blessings that have overflowed in our lives.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eGratitude is one of the greatest fruits of piety. A pious believer realizes that all the blessings they experience—whether large or seemingly simple—flow purely from the generosity of Allah SWT, the Lord of all worlds.\u003c/p\u003e\n\u003cp\u003eFrom the moment we open our eyes in the morning, we are enveloped in an ocean of His blessings: the clean air we breathe for free, the heartbeat that continues without command, the eyes still able to behold the beauty of the world, and the tongue still granted the ability to mention His name. Yet, as humans, we often find it easier to remember a single small hardship than to ponder the thousands of blessings that have overflowed in our lives.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-virtue-of-gratitude-and-allahs-promise\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-virtue-of-gratitude-and-allahs-promise\" class=\"header-mark\" aria-label=\"Direct link to section: The Virtue of Gratitude and Allah\u0026rsquo;s Promise\"\u003e\u003c/a\u003eThe Virtue of Gratitude and Allah\u0026rsquo;s Promise\u003c/h2\u003e\u003cp\u003eIn the Qur\u0026rsquo;an, Allah SWT provides a grand promise for His grateful servants. Allah SWT says in Surah Ibrahim, verse 7:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eلَئِنْ شَكَرْتُمْ لَأَزِيدَنَّكُمْ ۖ وَلَئِنْ كَفَرْتُمْ إِنَّ عَذَابِي لَشَدِيدٌ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And [remember] when your Lord proclaimed, \u0026lsquo;If you are grateful, I will surely increase you [in favor]; but if you show ingratitude, indeed, My punishment is severe.\u0026rsquo;\u0026rdquo;\u003c/em\u003e (QS. Ibrahim: 7)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe increase of blessings promised here is not merely about an increase in material wealth or worldly achievements. More than that, it is a promise of \u003cstrong\u003ebarakah (blessings)\u003c/strong\u003e and \u003cstrong\u003econtentment of the heart\u003c/strong\u003e. Many people possess abundant wealth yet live in narrowness and anxiety. On the contrary, there are those who live in simplicity, yet their hearts are at peace, their families are tranquil, and their worship feels sweet. That is the essence of true increase in favor.\u003c/p\u003e\n\u003cp\u003eDespite Allah\u0026rsquo;s infinite blessings, He reminds us that very few humans actually fulfill this gratitude:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eوَقَلِيلٌ مِّنْ عِبَادِيَ الشَّكُورُ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And few of My servants are grateful.\u0026rdquo;\u003c/em\u003e (QS. Saba\u0026rsquo;: 13)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"emulating-the-gratitude-of-prophet-muhammad-saw\" class=\"headerLink\"\u003e\n \u003ca href=\"#emulating-the-gratitude-of-prophet-muhammad-saw\" class=\"header-mark\" aria-label=\"Direct link to section: Emulating the Gratitude of Prophet Muhammad SAW\"\u003e\u003c/a\u003eEmulating the Gratitude of Prophet Muhammad SAW\u003c/h2\u003e\u003cp\u003eIf we wish to learn the true nature of gratitude, our greatest exemplar is Prophet Muhammad (peace and blessings be upon him). He was a Prophet whose entry into Paradise was guaranteed and whose past and future shortcomings were forgiven. Yet, this noble guarantee did not make him complacent.\u003c/p\u003e\n\u003cp\u003eOn the contrary, the Prophet SAW prayed so long during the night that his feet would swell. When Aisyah (may Allah be pleased with her) asked in amazement, \u003cem\u003e\u0026ldquo;Why do you do this, O Messenger of Allah, when Allah has forgiven your past and future sins?\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe Prophet SAW replied beautifully:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eأَفَلَا أَكُونُ عَبْدًا شَكُورًا\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Should I not be a grateful servant?\u0026rdquo;\u003c/em\u003e (HR. Bukhari and Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eHe did not worship out of fear of punishment or solely in pursuit of rewards; rather, he was driven by a deep sense of love and gratitude in his chest. He also taught his Ummah to recognize daily blessings through supplications that begin with praise to Allah, such as:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eUpon Waking Up:\u003c/strong\u003e\n\u003cblockquote\u003e\n \u003cp\u003eاَلْحَمْدُ لِلَّهِ الَّذِيْ أَحْيَانَا بَعْدَ مَا أَمَاتَنَا وَإِلَيْهِ النُّشُوْرُ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Praise be to Allah Who gave us life after He had caused us to die (sleep), and to Him is the resurrection.\u0026rdquo;\u003c/em\u003e (HR. Bukhari)\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAfter Eating:\u003c/strong\u003e\n\u003cblockquote\u003e\n \u003cp\u003eالْحَمْدُ لِلَّهِ الَّذِي أَطْعَمَنِي هَذَا وَرَزَقَنِيهِ مِنْ غَيْرِ حَوْلٍ مِنِّي وَلَا قُوَّةٍ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Praise be to Allah Who has fed me this and provided it for me without any strength or power on my part, then his past sins will be forgiven.\u0026rdquo;\u003c/em\u003e (HR. Abu Dawud \u0026amp; Tirmidzi)\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eUpon Dressing:\u003c/strong\u003e\n\u003cblockquote\u003e\n \u003cp\u003eالْحَمْدُ لِلَّهِ الَّذِيْ كَسَانِيْ هَذَا الثَّوْبَ وَرَزَقَنِيْهِ مِنْ غَيْرِ حَوْلٍ مِنِّيْ وَلَا قُوَّةٍ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Praise be to Allah Who has clothed me with this garment and provided it for me without any power or strength on my part, then his past sins will be forgiven.\u0026rdquo;\u003c/em\u003e (HR. Abu Dawud \u0026amp; Tirmidzi)\u003c/p\u003e\n\u003c/blockquote\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eFor a believer, gratitude is not a seasonal act of worship, but the very breath that accompanies every activity of life.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-three-pillars-of-gratitude\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-three-pillars-of-gratitude\" class=\"header-mark\" aria-label=\"Direct link to section: The Three Pillars of Gratitude\"\u003e\u003c/a\u003eThe Three Pillars of Gratitude\u003c/h2\u003e\u003cp\u003eScholars explain that true gratitude (to become \u003cem\u003eAbdan Syakuran\u003c/em\u003e—a truly grateful servant) must fulfill three interconnected pillars:\u003c/p\u003e\n\u003ch3 id=\"1-gratitude-with-the-heart-al-qalb\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-gratitude-with-the-heart-al-qalb\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Gratitude with the Heart (Al-Qalb)\"\u003e\u003c/a\u003e1. Gratitude with the Heart (Al-Qalb)\u003c/h3\u003e\u003cp\u003eThe heart must honestly acknowledge and believe that every blessing we receive comes purely from Allah SWT. We must cleanse our hearts of arrogance and the belief that our success is solely due to our own intelligence, skills, or hard work.\u003c/p\u003e\n\u003ch3 id=\"2-gratitude-with-the-tongue-al-lisan\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-gratitude-with-the-tongue-al-lisan\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Gratitude with the Tongue (Al-Lisan)\"\u003e\u003c/a\u003e2. Gratitude with the Tongue (Al-Lisan)\u003c/h3\u003e\u003cp\u003eThe tongue should constantly remain moist with praises of Allah by reciting \u003cem\u003etahmid\u003c/em\u003e (Alhamdulillah) and mentioning His favors with the intention of glorifying the Bestower of Blessings (\u003cem\u003etahadduth bin-ni\u0026rsquo;mah\u003c/em\u003e), not for showing off or pride.\u003c/p\u003e\n\u003ch3 id=\"3-gratitude-with-the-limbs-al-jawarih\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-gratitude-with-the-limbs-al-jawarih\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Gratitude with the Limbs (Al-Jawarih)\"\u003e\u003c/a\u003e3. Gratitude with the Limbs (Al-Jawarih)\u003c/h3\u003e\u003cp\u003eThis pillar is the practical proof of our gratitude. It manifests in using every blessing Allah has given us as a tool to increase our obedience to Him:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eThe Blessing of Wealth:\u003c/strong\u003e Used for charity, helping others, and keeping it free from what is forbidden.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe Blessing of Knowledge:\u003c/strong\u003e Implemented sincerely and taught to guide others without arrogance.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe Blessing of Health:\u003c/strong\u003e Used to walk to the mosque, perform acts of worship, and do good deeds.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe Blessing of Free Time:\u003c/strong\u003e Filled with productive actions, seeking knowledge, and remembering Allah.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe Blessing of Authority/Status:\u003c/strong\u003e Used to establish justice and ease the affairs of the community.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe Blessing of Family:\u003c/strong\u003e Nurtured with patience so they grow into a righteous and obedient generation.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe Blessing of Senses (Eyes, Ears, Tongue):\u003c/strong\u003e Guarded from what Allah has prohibited.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eWithout the proof from our limbs, our gratitude is incomplete. We would only be \u0026ldquo;eloquent in speech\u0026rdquo; but not yet \u003cem\u003eAbdan Syakuran\u003c/em\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-two-most-deceiving-blessings\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-two-most-deceiving-blessings\" class=\"header-mark\" aria-label=\"Direct link to section: The Two Most Deceiving Blessings\"\u003e\u003c/a\u003eThe Two Most Deceiving Blessings\u003c/h2\u003e\u003cp\u003eThe Prophet SAW specifically warned us about two blessings that most often lead humans to negligence and loss:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eنِعْمَتَانِ مَغْبُوْنٌ فِيْهِمَا كَثِيْرٌ مِنَ النَّاسِ: الصِّحَّةُ وَالْفَرَاغُ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;There are two blessings which many people lose: health and free time.\u0026rdquo;\u003c/em\u003e (HR. Bukhari)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eOften, we only realize the value of health when we are lying sick in a hospital bed. We only realize the preciousness of free time when busyness or death knocks at our door.\u003c/p\u003e\n\u003cp\u003eWe must also remember that every blessing entrusted to us will be questioned on the Day of Judgment:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eثُمَّ لَتُسْأَلُنَّ يَوْمَئِذٍ عَنِ النَّعِيمِ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Then you will surely be asked that Day about the pleasures [you enjoyed].\u0026rdquo;\u003c/em\u003e (QS. At-Takatsur: 8)\u003c/p\u003e\n\u003c/blockquote\u003e\u003chr\u003e\n\u003ch2 id=\"beware-of-the-qarun-syndrome\" class=\"headerLink\"\u003e\n \u003ca href=\"#beware-of-the-qarun-syndrome\" class=\"header-mark\" aria-label=\"Direct link to section: Beware of the Qarun Syndrome\"\u003e\u003c/a\u003eBeware of the Qarun Syndrome\u003c/h2\u003e\u003cp\u003eWhy do humans easily neglect gratitude? The primary cause is that the heart becomes too focused on oneself, forgetting the Giver of Blessings. This is the very disease that destroyed Qarun of old. When he was reminded to be grateful for his wealth, he arrogantly replied:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eإِنَّمَا أُوْتِيْتُهُ عَلَىٰ عِلْمٍ عِنْدِيْ\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;I was only given it because of knowledge I have.\u0026rdquo;\u003c/em\u003e (QS. Al-Qashash: 78)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003ePerhaps we have never said this out loud, but there might be a subtle whisper in our hearts when success is achieved: \u003cem\u003e\u0026ldquo;This is thanks to my greatness, my strategy, my hard work.\u0026rdquo;\u003c/em\u003e We forget that the intellect to think, the health to work, and the opportunity to strive all come from Allah.\u003c/p\u003e\n\u003cp\u003eLet us change our perspective on the world: \u003cstrong\u003edo not feel ownership, but feel guardianship (feel entrusted)\u003c/strong\u003e. For whatever is in our hands today will one day leave us and become the subject of questioning before Allah SWT.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"physical-and-spiritual-blessings\" class=\"headerLink\"\u003e\n \u003ca href=\"#physical-and-spiritual-blessings\" class=\"header-mark\" aria-label=\"Direct link to section: Physical and Spiritual Blessings\"\u003e\u003c/a\u003ePhysical and Spiritual Blessings\u003c/h2\u003e\u003cp\u003eAllah SWT says in Surah Luqman, verse 20:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eأَلَمْ تَرَوْا أَنَّ اللَّهَ سَخَّرَ لَكُمْ مَا فِي السَّمَاوَاتِ وَمَا فِي الْأَرْضِ وَأَسْبَغَ عَلَيْكُمْ نِعَمَهُ ظَاهِرَةً وَبَاطِنَةً\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Do you not see that Allah has made subject to you whatever is in the heavens and whatever is in the earth and amply bestowed upon you His favors, both apparent and unapparent?\u0026rdquo;\u003c/em\u003e (QS. Luqman: 20)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eApparent blessings (\u003cem\u003ezahir\u003c/em\u003e) are easy to recognize: physical health, financial comfort, and the warmth of family. Yet, unapparent blessings (\u003cem\u003ebatin\u003c/em\u003e) are often far greater in value: the blessing of faith and Islam, the guidance that directs our steps, our sins that are covered by Allah from the eyes of others, and the disasters that are averted from us without our ever knowing.\u003c/p\u003e\n\u003cp\u003eLet us not wait for sickness to appreciate health. Let us not wait for hardship to appreciate ease. And let us not wait for separation or death to realize that every second of life is a priceless gift from Allah SWT.\u003c/p\u003e\n\u003cp\u003eMay Allah SWT always guide our hearts to be among the few, those who are truly grateful. \u003cem\u003eAamiin Ya Rabbal \u0026lsquo;Alamin.\u003c/em\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eAdapted from the Friday Sermon by Ustadz Muhammad Faishal Fadhli (Dakwah.id).\u003c/em\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Mosaic of Love", "date_published" : "2026-07-16T11:12:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mozaik-cinta/", "url" : "https://www.aantriono.com/en/mozaik-cinta/", "summary": "Creating aesthetic documents in LaTeX is not limited to scientific articles or textbooks. By using creative packages like shapepar and pgfornament, we can arrange text or poetry into unique shapes, one of which is a heart shape.\nThis article reviews the complete LaTeX code to create a heart-shaped paragraph of lyrics titled My Heart, decorated with elegant classical ornaments above and below it.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eCreating aesthetic documents in LaTeX is not limited to scientific articles or textbooks. By using creative packages like \u003ccode\u003eshapepar\u003c/code\u003e and \u003ccode\u003epgfornament\u003c/code\u003e, we can arrange text or poetry into unique shapes, one of which is a heart shape.\u003c/p\u003e\n\u003cp\u003eThis article reviews the complete LaTeX code to create a heart-shaped paragraph of lyrics titled \u003cstrong\u003eMy Heart\u003c/strong\u003e, decorated with elegant classical ornaments above and below it.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eHeart-shaped text is perfect for greeting cards, poetry posters, or romantic gifts. In LaTeX, this can be achieved precisely using the \u003ccode\u003eshapepar\u003c/code\u003e package, which automatically adjusts line breaking algorithms to fit the text inside a heart contour perfectly.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"complete-latex-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-latex-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete LaTeX Code\"\u003e\u003c/a\u003eComplete LaTeX Code\u003c/h2\u003e\u003cp\u003eHere is the complete LaTeX code used to generate the heart-shaped poem layout with a custom header and ornaments:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv 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4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan 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class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%\\pagestyle{empty}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eMaroon\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%\\begin{center}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[every node/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003einner sep=0pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[text width=14cm,align=center]\u003c/span\u003e(Text)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"c\"\u003e%\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [Blue,font=\u003cspan class=\"k\"\u003e\\large\\bfseries\u003c/span\u003e] at (0,-0.4) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\LARGE\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eMy Heart\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [Maroon,font=\u003cspan class=\"k\"\u003e\\large\\bfseries\u003c/span\u003e] at (0,-0.8) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Large\u003c/span\u003e\u003cspan class=\"nb\"\u003e{}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [anchor=south] at (0,0.5) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=3cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e69\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [anchor=north] at (0,-1.2) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=3cm,symmetry=c]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e69\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\par\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\bigskip\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\it\\large\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fontfamily\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epzc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\itshape\u003c/span\u003e \u003cspan class=\"k\"\u003e\\selectfont\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\medskip\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\heartpar\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\emph\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Large\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eDisini kau dan aku.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eTerbiasa bersama.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMenjalani kasih sayang.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eBahagia kudenganmu.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ePernahkah kau menguntai.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eHari paling indah.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eKu ukir nama kita berdua.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eDi sini surga kita.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eBila kita mencintai yang lain.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMungkinkah hati ini akan tegar.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSebisa mungkin tak akan pernah.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSayangku akan hilang.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eIf you love somebody.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eCould we be this strong.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eI will fight to win.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eOur love will conquer all.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eWouldn\u0026#39;t risk my love.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eEven just one night.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eOur love will stay in my heart.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMy heart.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ePernahkah kau menguntai.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eHari paling indah.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eKu ukir nama kita berdua.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eDisini surga kita.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eBila kita mencintai yg lain.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMungkin kah hati ini akan tegar.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSebisa mungkin tak akan pernah.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSayangku akan hilang.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eBila kita mencintai yang lain.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMungkinkah hati ini akan tegar.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSebisa mungkin tak akan pernah.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSayang ku akan hilang.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eIf you love somebody.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eCould we be this strong.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eI will fight to win.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eOur love will conquer all.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eWouldn\u0026#39;t risk my love.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eEven just one night.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eOur love will stay in my heart.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMy heart my heart.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMy heart. \u003cspan class=\"nb\"\u003e}}}}}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e ;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\thispagestyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eempty\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003chr\u003e\n\u003cp\u003eThe output of the code is as follows:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='/mozaik-cinta/output-code.webp' srcset=\"/mozaik-cinta/output-code_hu_3c43e4b4561a7379.webp 800w, /mozaik-cinta/output-code_hu_84ccbcf32376335b.webp 1200w, /mozaik-cinta/output-code_hu_de982237aa275edc.webp 1600w\" alt=\"Mosaic of Love Code Output\" height=\"632\" width=\"467\"\u003e\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"detailed-code-explanation\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-code-explanation\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Code Explanation\"\u003e\u003c/a\u003eDetailed Code Explanation\u003c/h2\u003e\u003cp\u003eLet\u0026rsquo;s dissect the key parts of the LaTeX code:\u003c/p\u003e\n\u003ch3 id=\"1-document-class-and-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-document-class-and-packages\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Document Class and Packages\"\u003e\u003c/a\u003e1. Document Class and Packages\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\documentclass[]{standalone}\u003c/code\u003e\u003c/strong\u003e\u003cbr\u003e\nThe standalone class generates a PDF page cropped precisely to the size of the elements drawn (the TikZ picture).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\usepackage{shapepar}\u003c/code\u003e\u003c/strong\u003e\u003cbr\u003e\nThis provides the core command \u003ccode\u003e\\heartpar\u003c/code\u003e which reshapes the paragraph boundary into a heart.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\usepackage[object=vectorian]{pgfornament}\u003c/code\u003e\u003c/strong\u003e\u003cbr\u003e\nLoads beautiful Victorian vector ornaments to decorate the title.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003e\u003ccode\u003e\\usepackage[usenames,svgnames,dvipsnames]{xcolor}\u003c/code\u003e\u003c/strong\u003e\u003cbr\u003e\nDefines standard and custom named colors (e.g. \u003ccode\u003eMaroon\u003c/code\u003e and \u003ccode\u003eBlue\u003c/code\u003e).\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-font-and-color-customization\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-font-and-color-customization\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Font and Color Customization\"\u003e\u003c/a\u003e2. Font and Color Customization\u003c/h3\u003e\u003cp\u003eInside the poem paragraph, we see:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fontfamily\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epzc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\itshape\u003c/span\u003e \u003cspan class=\"k\"\u003e\\selectfont\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cul\u003e\n\u003cli\u003eThis switches the typeface to \u003cstrong\u003eZapf Chancery\u003c/strong\u003e (\u003ccode\u003epzc\u003c/code\u003e), which is an elegant, calligraphy-style typeface. It adds a sophisticated and romantic feel to the text.\u003c/li\u003e\n\u003cli\u003eThe colors are coordinated using a deep \u003ccode\u003eMaroon\u003c/code\u003e for the outer/top structure and \u003ccode\u003eBlue\u003c/code\u003e for the main header and heart text.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-formatting-the-heart-paragraph-heartpar\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-formatting-the-heart-paragraph-heartpar\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Formatting the Heart Paragraph (\\heartpar)\"\u003e\u003c/a\u003e3. Formatting the Heart Paragraph (\u003ccode\u003e\\heartpar\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eThe lirik or poem is wrapped in \u003ccode\u003e\\heartpar{...}\u003c/code\u003e.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eDo not use manual line breaks (\u003ccode\u003e\\\\\u003c/code\u003e) inside \u003ccode\u003e\\heartpar\u003c/code\u003e, as this breaks the line-breaking algorithm that maps words into the heart layout.\u003c/li\u003e\n\u003cli\u003eIf the paragraph is too short, the shape will look distorted. If it is too long, it might overflow. Adjust font size or text length accordingly.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"4-vector-ornaments-pgfornament\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-vector-ornaments-pgfornament\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Vector Ornaments (pgfornament)\"\u003e\u003c/a\u003e4. Vector Ornaments (\u003ccode\u003epgfornament\u003c/code\u003e)\u003c/h3\u003e\u003cp\u003eWe use ornament \u003ccode\u003e69\u003c/code\u003e above and below the title:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=3cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e69\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=3cm,symmetry=c]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e69\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ccode\u003esymmetry=c\u003c/code\u003e applies a central symmetry rotation to the bottom ornament to mirror the top ornament perfectly.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"how-to-compile-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-compile-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: How to Compile the Document\"\u003e\u003c/a\u003eHow to Compile the Document\u003c/h2\u003e\u003cp\u003eTo compile this LaTeX document:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eCopy the code above and save it as \u003ccode\u003emosaic-love.tex\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eEnsure \u003ccode\u003eshapepar\u003c/code\u003e and \u003ccode\u003epgfornament\u003c/code\u003e packages are installed in your TeX distribution (or use an online platform like Overleaf where they are pre-installed).\u003c/li\u003e\n\u003cli\u003eCompile using the \u003cstrong\u003epdfLaTeX\u003c/strong\u003e or \u003cstrong\u003eXeLaTeX\u003c/strong\u003e engine.\u003c/li\u003e\n\u003cli\u003eThe output will be a single-page PDF containing the beautiful layout.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eCombining \u003ccode\u003eshapepar\u003c/code\u003e and \u003ccode\u003epgfornament\u003c/code\u003e within TikZ unlocks highly artistic typesetting possibilities. You can easily replace the text with your own lyrics, quotes, or cards. Happy LaTeXing!\u003c/p\u003e", "language": "en" }, { "title" : "Getting to Know Linux Mint: History, Installation Guide, and Usage", "date_published" : "2026-07-08T07:35:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/linux-mint/", "url" : "https://www.aantriono.com/en/linux-mint/", "summary": " IntroductionFor computer users looking to migrate from Windows or macOS to the open-source world, Linux Mint is almost always the top recommendation. Known for its familiar interface, outstanding stability, and exceptional out-of-the-box ease of use, Linux Mint has secured its place as one of the most popular desktop Linux distributions (distros) in the world.\nThis article provides a deep dive into the history and development of Linux Mint, a step-by-step installation guide, and basic usage instructions to help you get the most out of this operating system for your daily needs.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eFor computer users looking to migrate from Windows or macOS to the open-source world, \u003cstrong\u003eLinux Mint\u003c/strong\u003e is almost always the top recommendation. Known for its familiar interface, outstanding stability, and exceptional out-of-the-box ease of use, Linux Mint has secured its place as one of the most popular desktop Linux distributions (distros) in the world.\u003c/p\u003e\n\u003cp\u003eThis article provides a deep dive into the history and development of Linux Mint, a step-by-step installation guide, and basic usage instructions to help you get the most out of this operating system for your daily needs.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-history-and-development-of-linux-mint\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-history-and-development-of-linux-mint\" class=\"header-mark\" aria-label=\"Direct link to section: The History and Development of Linux Mint\"\u003e\u003c/a\u003eThe History and Development of Linux Mint\u003c/h2\u003e\u003cp\u003eLinux Mint was first released in \u003cstrong\u003e2006\u003c/strong\u003e by its founder, \u003cstrong\u003eClement Lefebvre\u003c/strong\u003e (affectionately known as Clem). Its initial release, version 1.0 code-named \u0026ldquo;Ada\u0026rdquo;, was a beta release based on Kubuntu.\u003c/p\u003e\n\u003ch3 id=\"early-days-and-core-philosophy\" class=\"headerLink\"\u003e\n \u003ca href=\"#early-days-and-core-philosophy\" class=\"header-mark\" aria-label=\"Direct link to section: Early Days and Core Philosophy\"\u003e\u003c/a\u003eEarly Days and Core Philosophy\u003c/h3\u003e\u003cp\u003eFrom its inception, Linux Mint\u0026rsquo;s philosophy has been clear: \u003cstrong\u003e\u0026ldquo;From freedom came elegance.\u0026rdquo;\u003c/strong\u003e Clem aimed to build a desktop operating system that was not only free and open source but also incredibly comfortable for average users to use without requiring complex technical configurations.\u003c/p\u003e\n\u003cp\u003eStarting with version 2.0 \u0026ldquo;Barbara\u0026rdquo;, Linux Mint transitioned its codebase to Ubuntu. This move was a massive success, as it leveraged Ubuntu\u0026rsquo;s vast package repositories while offering a modified GNOME 2 interface that felt familiar to Windows users (featuring a search-capable application menu in the bottom-left corner).\u003c/p\u003e\n\u003ch3 id=\"transition-period-and-the-birth-of-cinnamon\" class=\"headerLink\"\u003e\n \u003ca href=\"#transition-period-and-the-birth-of-cinnamon\" class=\"header-mark\" aria-label=\"Direct link to section: Transition Period and the Birth of Cinnamon\"\u003e\u003c/a\u003eTransition Period and the Birth of Cinnamon\u003c/h3\u003e\u003cp\u003eWhen GNOME released GNOME 3 in 2011 with a radical interface redesign that was poorly received by traditional users, the Linux Mint team took a bold step. In 2012, they introduced \u003cstrong\u003eCinnamon\u003c/strong\u003e, a custom desktop environment that maintained a traditional panel-based layout while utilizing modern under-the-hood technologies. Cinnamon has since become the defining flagship desktop of Linux Mint.\u003c/p\u003e\n\u003cp\u003eSince the release of \u003cstrong\u003eLinux Mint 17 \u0026ldquo;Qiana\u0026rdquo; (2014)\u003c/strong\u003e, the development team decided to base all future main releases exclusively on \u003cstrong\u003eUbuntu LTS (Long Term Support)\u003c/strong\u003e releases. This decision guarantees exceptional long-term stability and security patches for its users.\u003c/p\u003e\n\u003cp\u003eAs a contingency plan in case anything ever happens to Ubuntu, Linux Mint also maintains \u003cstrong\u003eLMDE (Linux Mint Debian Edition)\u003c/strong\u003e, which is based directly on Debian Stable. The latest stable version of this alternative edition is \u003cstrong\u003eLMDE 6 \u0026ldquo;Faye\u0026rdquo;\u003c/strong\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"linux-mint-desktop-editions\" class=\"headerLink\"\u003e\n \u003ca href=\"#linux-mint-desktop-editions\" class=\"header-mark\" aria-label=\"Direct link to section: Linux Mint Desktop Editions\"\u003e\u003c/a\u003eLinux Mint Desktop Editions\u003c/h2\u003e\u003cp\u003eLinux Mint officially offers three distinct desktop environments:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eCinnamon:\u003c/strong\u003e The flagship and most modern edition. It offers advanced visual effects, full customization options, and an elegant classic interface.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMATE:\u003c/strong\u003e A continuation of the legendary GNOME 2 desktop. It is lighter than Cinnamon and highly stable, making it ideal for mid-range computers.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eXfce:\u003c/strong\u003e The most lightweight and resource-friendly option. It is extremely stable and specifically designed to revive older computers and laptops.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eAs of 2026, the latest stable release is \u003cstrong\u003eLinux Mint 22.3 \u0026ldquo;Zena\u0026rdquo;\u003c/strong\u003e, which is based on Ubuntu 24.04 LTS and receives official support until April 2029.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"step-by-step-linux-mint-installation-guide\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-linux-mint-installation-guide\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step Linux Mint Installation Guide\"\u003e\u003c/a\u003eStep-by-Step Linux Mint Installation Guide\u003c/h2\u003e\u003cp\u003eInstalling Linux Mint is straightforward thanks to its intuitive graphical installer. Here is how to prepare and carry out the installation:\u003c/p\u003e\n\u003ch3 id=\"1-preparation\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-preparation\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Preparation\"\u003e\u003c/a\u003e1. Preparation\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eDownload the ISO File:\u003c/strong\u003e Download the official Linux Mint ISO file (Cinnamon, MATE, or Xfce edition) from \u003ca href=\"https://linuxmint.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003elinuxmint.com\u003c/a\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCreate a Bootable USB:\u003c/strong\u003e Use software like Rufus (on Windows) or BalenaEtcher (on macOS/Linux) to write the ISO to a USB flash drive (minimum 4GB capacity).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBackup Your Data:\u003c/strong\u003e Always back up your important files to external storage before starting the installation process.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-booting-from-the-live-usb\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-booting-from-the-live-usb\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Booting from the Live USB\"\u003e\u003c/a\u003e2. Booting from the Live USB\u003c/h3\u003e\u003cul\u003e\n\u003cli\u003eInsert the bootable USB flash drive into your computer.\u003c/li\u003e\n\u003cli\u003eRestart your computer and access the BIOS/UEFI boot menu (usually by pressing \u003ccode\u003eF2\u003c/code\u003e, \u003ccode\u003eF12\u003c/code\u003e, \u003ccode\u003eDel\u003c/code\u003e, or \u003ccode\u003eEsc\u003c/code\u003e during startup).\u003c/li\u003e\n\u003cli\u003eChange the boot order to prioritize the USB drive.\u003c/li\u003e\n\u003cli\u003eSave changes and let the system boot into the Linux Mint \u0026ldquo;Live Desktop\u0026rdquo; environment.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-the-installation-process\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-the-installation-process\" class=\"header-mark\" aria-label=\"Direct link to section: 3. The Installation Process\"\u003e\u003c/a\u003e3. The Installation Process\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eOn the Live Desktop, double-click the \u003cstrong\u003e\u0026ldquo;Install Linux Mint\u0026rdquo;\u003c/strong\u003e icon.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eChoose Language:\u003c/strong\u003e Select your preferred installation language and click \u003cem\u003eContinue\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eKeyboard Layout:\u003c/strong\u003e Choose your keyboard layout (usually the default \u003cem\u003eEnglish (US)\u003c/em\u003e).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eInstall Multimedia Codecs:\u003c/strong\u003e Check the box that says \u003cem\u003e“Install multimedia codecs”\u003c/em\u003e so you can play popular video and audio formats immediately.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eInstallation Type:\u003c/strong\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eErase disk and install Linux Mint:\u003c/strong\u003e Select this to wipe the entire drive and use it solely for Linux Mint.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSomething else:\u003c/strong\u003e Select this if you want to partition the drive manually or configure a \u003cem\u003eDual-Boot\u003c/em\u003e setup alongside Windows.\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSelect Time Zone:\u003c/strong\u003e Choose your location on the map to set the system clock.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCreate User Account:\u003c/strong\u003e Enter your name, computer name, username, and password. Make sure to remember this password, as it will be required for administrative tasks.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFile Copying:\u003c/strong\u003e Wait for the installation progress bar to complete (usually takes 10 to 20 minutes depending on storage speeds).\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFinish:\u003c/strong\u003e When the installation is complete, click \u003cem\u003eRestart Now\u003c/em\u003e and remove the USB drive when prompted.\u003c/li\u003e\n\u003c/ol\u003e\n\u003chr\u003e\n\u003ch2 id=\"basic-linux-mint-usage-guide\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-linux-mint-usage-guide\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Linux Mint Usage Guide\"\u003e\u003c/a\u003eBasic Linux Mint Usage Guide\u003c/h2\u003e\u003cp\u003eOnce your computer restarts, you will be greeted by a \u003cstrong\u003eWelcome Screen\u003c/strong\u003e that guides you through your initial setup. Here are the core tools you should know:\u003c/p\u003e\n\u003ch3 id=\"1-updating-your-system-update-manager\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-updating-your-system-update-manager\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Updating Your System (Update Manager)\"\u003e\u003c/a\u003e1. Updating Your System (Update Manager)\u003c/h3\u003e\u003cp\u003eThe first thing you should do after installing is to ensure your system is secure. Open the \u003cstrong\u003eUpdate Manager\u003c/strong\u003e (shield icon in the system tray), click \u003cem\u003eRefresh\u003c/em\u003e, and then \u003cem\u003eInstall Updates\u003c/em\u003e. Updates in Linux Mint are carefully categorized for safety, meaning they are highly unlikely to break your system configuration.\u003c/p\u003e\n\u003ch3 id=\"2-installing-applications-software-manager\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-installing-applications-software-manager\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Installing Applications (Software Manager)\"\u003e\u003c/a\u003e2. Installing Applications (Software Manager)\u003c/h3\u003e\u003cp\u003eThere is no need to search for installer files online. Simply open the \u003cstrong\u003eSoftware Manager\u003c/strong\u003e, which acts like a built-in App Store. You will find thousands of popular applications like LibreOffice, GIMP, VLC, Steam, and Spotify available to install with a single click. It also natively supports \u003cstrong\u003eFlatpak\u003c/strong\u003e packages to give you access to the latest software versions.\u003c/p\u003e\n\u003ch3 id=\"3-system-restore-via-timeshift\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-system-restore-via-timeshift\" class=\"header-mark\" aria-label=\"Direct link to section: 3. System Restore via Timeshift\"\u003e\u003c/a\u003e3. System Restore via Timeshift\u003c/h3\u003e\u003cp\u003eTimeshift comes pre-installed in Linux Mint. It automatically takes snapshots of your system files. If your system ever behaves unexpectedly after a driver update or a manual configuration mistake, you can restore it to a previous working state in a matter of minutes.\u003c/p\u003e\n\u003ch3 id=\"4-desktop-customization\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-desktop-customization\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Desktop Customization\"\u003e\u003c/a\u003e4. Desktop Customization\u003c/h3\u003e\u003cp\u003eIn the Cinnamon edition, you can easily change the desktop theme, icons, window borders, and add desktop widgets (\u003cem\u003eDesklets\u003c/em\u003e). Navigate to \u003cstrong\u003eSystem Settings\u003c/strong\u003e to explore the highly flexible customization options.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eLinux Mint proves that using a Linux operating system does not have to be intimidating or restricted to typing commands in a terminal. With its strong focus on user convenience, long-term stability backed by Ubuntu LTS, and a welcoming community, Linux Mint is one of the best daily operating systems for both beginners and professionals who value a stable, productive environment.\u003c/p\u003e", "language": "en" }, { "title" : "The Danger of Arrogance: A Barrier to Paradise and Destroyer of Deeds", "date_published" : "2026-07-08T07:04:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bahaya-sifat-sombong/", "url" : "https://www.aantriono.com/en/bahaya-sifat-sombong/", "summary": " IntroductionOne of the most dangerous and subtle spiritual diseases that infects human hearts is arrogance (kibr). Pride often slips into a person\u0026rsquo;s heart unnoticed, whether due to their vast knowledge, noble lineage, physical beauty, or material wealth. This quality is deeply detested by Allah SWT, as arrogance is, in essence, rejecting the truth and looking down upon other creations of Allah.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eOne of the most dangerous and subtle spiritual diseases that infects human hearts is arrogance (\u003cem\u003ekibr\u003c/em\u003e). Pride often slips into a person\u0026rsquo;s heart unnoticed, whether due to their vast knowledge, noble lineage, physical beauty, or material wealth. This quality is deeply detested by Allah SWT, as arrogance is, in essence, rejecting the truth and looking down upon other creations of Allah.\u003c/p\u003e\n\u003ch2 id=\"scriptural-proofs-on-the-danger-of-arrogance\" class=\"headerLink\"\u003e\n \u003ca href=\"#scriptural-proofs-on-the-danger-of-arrogance\" class=\"header-mark\" aria-label=\"Direct link to section: Scriptural Proofs on the Danger of Arrogance\"\u003e\u003c/a\u003eScriptural Proofs on the Danger of Arrogance\u003c/h2\u003e\u003cp\u003eIn Islamic teachings, arrogance is an absolute barrier that prevents a person from experiencing the beauty of Paradise. Prophet Muhammad (PBUH) gave a stern warning regarding this:\u003c/p\u003e\n\u003cp\u003eلَا يَدْخُلُ الْجَنَّةَ مَنْ كَانَ فِي قَلْبِهِ مِثْقَالُ ذَرَّةٍ مِنْ كِبْرٍ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;He will not enter Paradise who has even a speck of arrogance in his heart.\u0026rdquo;\u003c/em\u003e (Narrated by Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eFurthermore, the Prophet (PBUH) defined the true meaning of arrogance so that we may guard ourselves against it:\u003c/p\u003e\n\u003cp\u003eالْكِبْرُ بَطَرُ الْحَقِّ وَغَمْطُ النَّاسِ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Arrogance is rejecting the truth and looking down on people.\u0026rdquo;\u003c/em\u003e (Narrated by Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"destructive-impacts-of-arrogance-on-the-self\" class=\"headerLink\"\u003e\n \u003ca href=\"#destructive-impacts-of-arrogance-on-the-self\" class=\"header-mark\" aria-label=\"Direct link to section: Destructive Impacts of Arrogance on the Self\"\u003e\u003c/a\u003eDestructive Impacts of Arrogance on the Self\u003c/h2\u003e\u003cp\u003eAllowing the seeds of pride to grow in the heart leads to destructive consequences for both our spiritual and social life:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eBeing Deprived of Allah\u0026rsquo;s Guidance:\u003c/strong\u003e Allah SWT promises that He will turn away the hearts of those who are arrogant from understanding His signs and truth.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDestroying the Value of Good Deeds:\u003c/strong\u003e No matter how great our worship is, if it is mixed with self-admiration (\u003cem\u003eujub\u003c/em\u003e) and arrogance, its reward will perish before Allah SWT.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTriggering Social Dislike:\u003c/strong\u003e An arrogant person who belittles others will naturally be shunned in social circles because of their unpleasant attitude.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLeading to Eternal Ruin:\u003c/strong\u003e History records how Iblis (Satan) was expelled from Paradise, and Pharaoh and Qarun were destroyed by Allah solely because of their arrogance.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe only One who holds the right to absolute pride and greatness is Allah SWT. Humans, who are created from dust and will eventually return to dust, have no reason to be arrogant. The best remedy for pride is to recognize our own weaknesses and constantly cultivate \u003cem\u003etawadhu\u003c/em\u003e (humility). May Allah SWT cleanse our hearts from all forms of arrogance, no matter how small.\u003c/p\u003e", "language": "en" }, { "title" : "The Virtue of Charity: An Eternal Investment That Purifies Wealth", "date_published" : "2026-07-08T07:03:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/keutamaan-sedekah/", "url" : "https://www.aantriono.com/en/keutamaan-sedekah/", "summary": " IntroductionMany people mistakenly believe that by sharing their wealth with others, their overall riches will decrease. This view is a significant error. In the eyes of faith, giving or spending in charity (sadaqah) is actually the best way to grow and purify one\u0026rsquo;s wealth. Charity not only provides essential social support for the receiver but also guarantees security and peace of mind for the giver.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eMany people mistakenly believe that by sharing their wealth with others, their overall riches will decrease. This view is a significant error. In the eyes of faith, giving or spending in charity (\u003cem\u003esadaqah\u003c/em\u003e) is actually the best way to grow and purify one\u0026rsquo;s wealth. Charity not only provides essential social support for the receiver but also guarantees security and peace of mind for the giver.\u003c/p\u003e\n\u003ch2 id=\"scriptural-proofs-on-the-virtue-of-charity\" class=\"headerLink\"\u003e\n \u003ca href=\"#scriptural-proofs-on-the-virtue-of-charity\" class=\"header-mark\" aria-label=\"Direct link to section: Scriptural Proofs on the Virtue of Charity\"\u003e\u003c/a\u003eScriptural Proofs on the Virtue of Charity\u003c/h2\u003e\u003cp\u003eAllah SWT describes charity as a fertile seed that yields abundant stalks of goodness. He says in the Holy Qur\u0026rsquo;an:\u003c/p\u003e\n\u003cp\u003eمَّثَلُ الَّذِينَ يُنفِقُونَ أَمْوَالَهُمْ فِي سَبِيلِ اللَّهِ كَمَثَلِ حَبَّةٍ أَنبَتَتْ سَبْعَ سَنَابِلَ فِي كُلِّ سُنبُلَةٍ مِّائَةُ حَبَّةٍ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;The example of those who spend their wealth in the way of Allah is like a seed of grain which grows seven spikes; in each spike is a hundred grains.\u0026rdquo;\u003c/em\u003e (QS. Al-Baqarah: 261)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eProphet Muhammad (PBUH) reinforced this with his oath that giving in charity does not reduce one\u0026rsquo;s wealth in the slightest:\u003c/p\u003e\n\u003cp\u003eمَا نَقَصَتْ صَدَقَةٌ مِنْ مَالٍ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Charity does not decrease wealth.\u0026rdquo;\u003c/em\u003e (Narrated by Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"wisdom-and-benefits-of-giving-charity\" class=\"headerLink\"\u003e\n \u003ca href=\"#wisdom-and-benefits-of-giving-charity\" class=\"header-mark\" aria-label=\"Direct link to section: Wisdom and Benefits of Giving Charity\"\u003e\u003c/a\u003eWisdom and Benefits of Giving Charity\u003c/h2\u003e\u003cp\u003eBehind the encouragement to give charity lie vast spiritual and humanitarian wisdoms:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eExpiation of Sins:\u003c/strong\u003e Sincere charity cools the anger of Allah SWT and expiates sins, just as water extinguishes fire.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePurifying Soul and Wealth:\u003c/strong\u003e Sadaqah cleanses the heart from greed, extreme love for worldly life, and purifies our assets from the rights of the weak that have not been delivered.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eShade on the Day of Judgment:\u003c/strong\u003e The wealth we give in charity will act as a protective canopy shading us on the Day of Resurrection, when the sun is brought close and there is no other shade.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eOpening New Doors of Sustenance:\u003c/strong\u003e Allah promises to replace whatever His servants spend with something far better and more blessed.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eCharity is not about how large of an amount we give, but rather about the sincerity of the heart and our care for one another. The true wealth we possess is not what we store and enjoy in this life, but what we send ahead through charity as our eternal provisions in the hereafter. Let us build a habit of giving charity, no matter how small, to purify our wealth and souls.\u003c/p\u003e", "language": "en" }, { "title" : "Maintaining Family Ties: A Bridge of Love and Key to Abundant Sustenance", "date_published" : "2026-07-08T07:02:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menjaga-silaturahmi/", "url" : "https://www.aantriono.com/en/menjaga-silaturahmi/", "summary": " IntroductionHumans are created as social beings who cannot live entirely on their own. We need each other to complement and support one another in times of need. In Islam, these social interactions are beautifully guided through the concept of silaturahmi—specifically maintaining family ties and fostering love among relatives. Silaturahmi is not just exchanging polite greetings; it is a sincere expression of care to support, help, and maintain harmony among relatives.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eHumans are created as social beings who cannot live entirely on their own. We need each other to complement and support one another in times of need. In Islam, these social interactions are beautifully guided through the concept of \u003cem\u003esilaturahmi\u003c/em\u003e—specifically maintaining family ties and fostering love among relatives. Silaturahmi is not just exchanging polite greetings; it is a sincere expression of care to support, help, and maintain harmony among relatives.\u003c/p\u003e\n\u003ch2 id=\"scriptural-proofs-on-maintaining-family-ties\" class=\"headerLink\"\u003e\n \u003ca href=\"#scriptural-proofs-on-maintaining-family-ties\" class=\"header-mark\" aria-label=\"Direct link to section: Scriptural Proofs on Maintaining Family Ties\"\u003e\u003c/a\u003eScriptural Proofs on Maintaining Family Ties\u003c/h2\u003e\u003cp\u003eMaintaining family ties is a religious duty that brings immense blessings. Prophet Muhammad (PBUH) stated that keeping family bonds is a primary cause for expanding one\u0026rsquo;s sustenance and prolonging one\u0026rsquo;s lifespan with blessings.\u003c/p\u003e\n\u003cp\u003eHe said:\u003c/p\u003e\n\u003cp\u003eمَنْ أَحَبَّ أَنْ يُبْسَطَ لَهُ فِي رِزْقِهِ، وَيُنْسَأَ لَهُ فِي أَثَرِهِ، فَلْيَصِلْ رَحِمَهُ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Whoever loves that his sustenance be expanded and his lifespan be prolonged, let him keep family ties.\u0026rdquo;\u003c/em\u003e (Narrated by Al-Bukhari \u0026amp; Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eConversely, Islam gives a stern warning to those who intentionally break off relations:\u003c/p\u003e\n\u003cp\u003eلَا يَدْخُلُ الْجَنَّةَ قَاطِعٌ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;The one who cuts (family ties) will not enter Paradise.\u0026rdquo;\u003c/em\u003e (Narrated by Al-Bukhari \u0026amp; Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"blessings-of-fostering-family-bonds\" class=\"headerLink\"\u003e\n \u003ca href=\"#blessings-of-fostering-family-bonds\" class=\"header-mark\" aria-label=\"Direct link to section: Blessings of Fostering Family Bonds\"\u003e\u003c/a\u003eBlessings of Fostering Family Bonds\u003c/h2\u003e\u003cp\u003eGuarding family connections brings broad positive impacts on both personal and social dimensions of life:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eExpanding Sustenance:\u003c/strong\u003e Healthy interactions with relatives often open doors to cooperation, financial assistance, or ease in business affairs.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eBlessings in Lifespan:\u003c/strong\u003e A prolonged lifespan in the context of the hadith is interpreted by scholars as a life full of blessings, utility, and a good legacy that remains even after death.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAttaining Allah\u0026rsquo;s Mercy:\u003c/strong\u003e Allah loves those who spread compassion. Fostering family ties connects us directly with the mercy of Allah SWT.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePeace of Mind:\u003c/strong\u003e Staying in harmony with family members protects us from the stress and exhausting conflicts of family estrangement.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eRebuilding family ties sometimes requires a great deal of humility, especially when relationships have been strained by misunderstandings. However, as believers seeking the pleasure of Allah and abundant blessings, we must take the first step to reconcile. True \u003cem\u003esilaturahmi\u003c/em\u003e is not merely returning the kindness of those who are good to us, but reconnecting with relatives who have drifted away or become distant.\u003c/p\u003e", "language": "en" }, { "title" : "The Beauty of Patience: A Pillar of Faith and Key to Victory", "date_published" : "2026-07-08T07:01:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/keindahan-sabar/", "url" : "https://www.aantriono.com/en/keindahan-sabar/", "summary": " IntroductionLife in this world is never free from ups and downs. At times we find ourselves experiencing happiness and prosperity, while at other times we are tested with difficulties, sorrow, or loss. This is where the virtue of patience (sabr) becomes vital. Patience does not mean passive surrender without effort; rather, it is restraining oneself from complaints, keeping emotions in check, and remaining steadfast on the path of truth when the storms of life hit. Patience is a guiding light for a believer.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eLife in this world is never free from ups and downs. At times we find ourselves experiencing happiness and prosperity, while at other times we are tested with difficulties, sorrow, or loss. This is where the virtue of patience (\u003cem\u003esabr\u003c/em\u003e) becomes vital. Patience does not mean passive surrender without effort; rather, it is restraining oneself from complaints, keeping emotions in check, and remaining steadfast on the path of truth when the storms of life hit. Patience is a guiding light for a believer.\u003c/p\u003e\n\u003ch2 id=\"scriptural-proofs-on-the-virtue-of-patience\" class=\"headerLink\"\u003e\n \u003ca href=\"#scriptural-proofs-on-the-virtue-of-patience\" class=\"header-mark\" aria-label=\"Direct link to section: Scriptural Proofs on the Virtue of Patience\"\u003e\u003c/a\u003eScriptural Proofs on the Virtue of Patience\u003c/h2\u003e\u003cp\u003eAllah SWT loves those who are patient. In the Holy Qur\u0026rsquo;an, Allah SWT repeatedly emphasizes the high status of patience:\u003c/p\u003e\n\u003cp\u003eإِنَّمَا يُوَفَّى الصَّابِرُونَ أَجْرَهُم بِغَيْرِ حِسَابٍ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Indeed, the patient will be given their reward without account.\u0026rdquo;\u003c/em\u003e (QS. Az-Zumar: 10)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eProphet Muhammad (PBUH) also gave profound advice regarding patience as the greatest gift Allah can bestow upon a servant:\u003c/p\u003e\n\u003cp\u003eوَمَا أُعْطِيَ أَحَدٌ عَطَاءً خَيْرًا وَأَوْسَعَ مِنَ الصَّبْرِ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;And no one is given a gift better and more comprehensive than patience.\u0026rdquo;\u003c/em\u003e (Narrated by Al-Bukhari \u0026amp; Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"three-dimensions-of-patience-in-islam\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-dimensions-of-patience-in-islam\" class=\"header-mark\" aria-label=\"Direct link to section: Three Dimensions of Patience in Islam\"\u003e\u003c/a\u003eThree Dimensions of Patience in Islam\u003c/h2\u003e\u003cp\u003eIslamic scholars categorize patience into three primary dimensions:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003ePatience in Obeying Allah:\u003c/strong\u003e Requiring consistency and inner struggle to perform acts of worship regularly, even when faced with fatigue or laziness.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePatience in Avoiding Sins:\u003c/strong\u003e Controlling one\u0026rsquo;s desires and resisting the temptation of forbidden things, which often appear appealing and pleasurable.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003ePatience in Accepting the Decrees of Allah:\u003c/strong\u003e Facing calamities, illness, or loss with contentment and a firm belief that there is divine wisdom behind every decree.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003ePatience is like medicine; it may taste bitter at first, but its healing effects are sweet and lasting. When we face every trial with beautiful patience (\u003cem\u003esabrun jamil\u003c/em\u003e), Allah promises not only peace of mind in this world but also limitless rewards in the hereafter. Let us take patience and prayer as our constants in navigating the ocean of life.\u003c/p\u003e", "language": "en" }, { "title" : "Guarding the Tongue: The Key to Salvation and Peace of Mind", "date_published" : "2026-07-08T07:00:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menjaga-lisan/", "url" : "https://www.aantriono.com/en/menjaga-lisan/", "summary": " IntroductionThe tongue is one of the greatest blessings Allah SWT has bestowed upon mankind. Through speech, we communicate, remember Allah, and spread goodness. However, the tongue can also become a destructive weapon if not guarded carefully. Many disputes, enmities, and even conflicts begin with a single, uncontrolled word. Therefore, guarding the tongue is an essential reflection of a Muslim\u0026rsquo;s faith.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThe tongue is one of the greatest blessings Allah SWT has bestowed upon mankind. Through speech, we communicate, remember Allah, and spread goodness. However, the tongue can also become a destructive weapon if not guarded carefully. Many disputes, enmities, and even conflicts begin with a single, uncontrolled word. Therefore, guarding the tongue is an essential reflection of a Muslim\u0026rsquo;s faith.\u003c/p\u003e\n\u003ch2 id=\"scriptural-proofs-on-guarding-the-tongue\" class=\"headerLink\"\u003e\n \u003ca href=\"#scriptural-proofs-on-guarding-the-tongue\" class=\"header-mark\" aria-label=\"Direct link to section: Scriptural Proofs on Guarding the Tongue\"\u003e\u003c/a\u003eScriptural Proofs on Guarding the Tongue\u003c/h2\u003e\u003cp\u003eIn Islamic teachings, guarding the tongue is highly emphasized. Prophet Muhammad (PBUH) directly linked the perfection of faith in Allah and the Last Day to the obligation of speaking good or remaining silent.\u003c/p\u003e\n\u003cp\u003eHe said:\u003c/p\u003e\n\u003cp\u003eمَنْ كَانَ يُؤْمِنُ بِاللَّهِ وَالْيَوْمِ الآخِرِ فَلْيَقُلْ خَيْرًا أَوْ لِيَصْمُتْ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Whoever believes in Allah and the Last Day, let him speak goodness or remain silent.\u0026rdquo;\u003c/em\u003e (Narrated by Al-Bukhari \u0026amp; Muslim)\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eFurthermore, Allah SWT warns in the Holy Qur\u0026rsquo;an that every human word is closely monitored by angels assigned to record deeds:\u003c/p\u003e\n\u003cp\u003eمَا يَلْفِظُ مِنْ قَوْلٍ إِلَّا لَدَيْهِ رَقِيبٌ عَتِيدٌ\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eMeaning:\u003c/strong\u003e \u003cem\u003e\u0026ldquo;Not a word does he utter but there is a watcher by him, ready (to record it).\u0026rdquo;\u003c/em\u003e (QS. Qaf: 18)\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"wisdom-behind-guarding-the-tongue\" class=\"headerLink\"\u003e\n \u003ca href=\"#wisdom-behind-guarding-the-tongue\" class=\"header-mark\" aria-label=\"Direct link to section: Wisdom Behind Guarding the Tongue\"\u003e\u003c/a\u003eWisdom Behind Guarding the Tongue\u003c/h2\u003e\u003cp\u003eWhen a believer disciplines themselves to filter every word they speak, they reap numerous spiritual and social benefits:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eAvoidance of Major Sins:\u003c/strong\u003e Many major sins originate from the tongue, such as backbiting (ghibah), slander, lying, and insults. Guarding the tongue closes the doors to these transgressions.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAttaining Allah\u0026rsquo;s Pleasure:\u003c/strong\u003e Speaking kind, reconciliatory, and beneficial words is a form of charity that earns the pleasure and reward of Allah SWT.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFostering Harmonious Social Relations:\u003c/strong\u003e Polite and respectful words strengthen brotherhood and minimize misunderstandings among people.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAchieving Inner Peace:\u003c/strong\u003e Those who guard their tongue enjoy greater tranquility of heart, as they are free from the burden of regret that follows careless speech.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eGuarding the tongue is not easy, especially in our digital age where our thumbs often act as a secondary tongue on social media. However, with the awareness that every word and written post will be accounted for before Allah SWT, we must constantly strive to train ourselves. Keep in mind the beautiful advice of the Prophet (PBUH): speak what is good and beneficial, or choose the peace of silence.\u003c/p\u003e", "language": "en" }, { "title" : "The Story of Leaving Falsehood: From Sinner to Pious Believer", "date_published" : "2026-07-08T06:53:00+07:00", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/meninggalkan-kebohongan/", "url" : "https://www.aantriono.com/en/meninggalkan-kebohongan/", "summary": " IntroductionMany people wish to distance themselves from bad habits and sins, yet they often find it incredibly difficult to break free. The story of a man who embraced Islam during the time of Prophet Muhammad (peace be upon him) shows that great transformations can begin with a single, simple commitment: guarding one\u0026rsquo;s honesty. Through a single piece of advice from the Prophet (PBUH), he succeeded in abandoning a lifetime of wrongdoing and embarking on a path of righteousness.\nThis exemplary story is documented in the book Kumpulan Kisah Teladan (A Collection of Exemplary Stories) compiled by Prof. Dr. H. M. Hasballah Thaib, M.A. and H. Zamakhsyari Hasballah, Lc., M.A., Ph.D., which references the work Qashash Muatssirah li al-Syabab by Iyyadh Faiz.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eMany people wish to distance themselves from bad habits and sins, yet they often find it incredibly difficult to break free. The story of a man who embraced Islam during the time of Prophet Muhammad (peace be upon him) shows that great transformations can begin with a single, simple commitment: guarding one\u0026rsquo;s honesty. Through a single piece of advice from the Prophet (PBUH), he succeeded in abandoning a lifetime of wrongdoing and embarking on a path of righteousness.\u003c/p\u003e\n\u003cp\u003eThis exemplary story is documented in the book \u003cem\u003eKumpulan Kisah Teladan\u003c/em\u003e (A Collection of Exemplary Stories) compiled by Prof. Dr. H. M. Hasballah Thaib, M.A. and H. Zamakhsyari Hasballah, Lc., M.A., Ph.D., which references the work \u003cem\u003eQashash Muatssirah li al-Syabab\u003c/em\u003e by Iyyadh Faiz.\u003c/p\u003e\n\u003ch2 id=\"the-prophets-simple-counsel\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-prophets-simple-counsel\" class=\"header-mark\" aria-label=\"Direct link to section: The Prophet\u0026rsquo;s Simple Counsel\"\u003e\u003c/a\u003eThe Prophet\u0026rsquo;s Simple Counsel\u003c/h2\u003e\u003cp\u003eOne day, a man came to Prophet Muhammad (PBUH) to embrace Islam. After reciting the testimony of faith (\u003cem\u003eshahada\u003c/em\u003e), he opened his heart and confessed his personal struggles:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u0026ldquo;O Messenger of Allah, I am accustomed to committing sins and find it very difficult to leave them.\u0026rdquo;\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eRather than demanding that the man immediately abandon all his vices, the Prophet (PBUH) made a single, straightforward request:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u0026ldquo;Will you promise me that you will never tell a lie?\u0026rdquo;\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe man readily replied, \u0026ldquo;Yes, I promise.\u0026rdquo; With that promise, the man departed and returned home.\u003c/p\u003e\n\u003ch2 id=\"the-internal-struggle\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-internal-struggle\" class=\"header-mark\" aria-label=\"Direct link to section: The Internal Struggle\"\u003e\u003c/a\u003eThe Internal Struggle\u003c/h2\u003e\u003cp\u003eBefore accepting Islam, the man had been notorious for stealing, gambling, and drinking alcohol. Although he sincerely wanted to reform his life after becoming a Muslim, the weight of his old habits felt overwhelming.\u003c/p\u003e\n\u003cp\u003eOn his way home, the Prophet’s words echoed in his mind. Whenever an opportunity or temptation to commit a sin arose, he was instantly reminded of his pledge to speak only the truth.\u003c/p\u003e\n\u003cp\u003eHe reasoned with himself: if the Prophet (PBUH) were to ask him about his actions, what would he say?\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eIf he lied to cover up his deeds, he would break his sacred promise.\u003c/li\u003e\n\u003cli\u003eBut if he spoke the truth, he would have to face the consequences and shame of admitting his crimes to the Messenger of Allah.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eCaught between the choice of breaking his word or exposing his sins, he chose to walk away from temptation. Every time he was on the verge of doing something wrong, he desisted out of the fear of having to admit it truthfully.\u003c/p\u003e\n\u003ch2 id=\"from-falsehood-to-righteousness\" class=\"headerLink\"\u003e\n \u003ca href=\"#from-falsehood-to-righteousness\" class=\"header-mark\" aria-label=\"Direct link to section: From Falsehood to Righteousness\"\u003e\u003c/a\u003eFrom Falsehood to Righteousness\u003c/h2\u003e\u003cp\u003eOver time, the man realized the profound wisdom behind the Prophet\u0026rsquo;s advice. By guarding his tongue from falsehood, he had clean built a shield against all other evils. The simple commitment to honesty naturally compelled him to abandon theft, gambling, and alcohol.\u003c/p\u003e\n\u003cp\u003eHe continued to master his desires until he was completely cured of his old vices. His life transformed completely; he abandoned the path of delinquency and became a devout, righteous believer of noble character.\u003c/p\u003e\n\u003ch2 id=\"the-ripple-effect-of-truthfulness\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-ripple-effect-of-truthfulness\" class=\"header-mark\" aria-label=\"Direct link to section: The Ripple Effect of Truthfulness\"\u003e\u003c/a\u003eThe Ripple Effect of Truthfulness\u003c/h2\u003e\u003cp\u003eThis story teaches us that honesty holds immense power over our actions. When a person commits to being truthful, they become far more mindful of their behavior. The reluctance to lie to cover up a mistake acts as a powerful deterrent against committing the mistake in the first place.\u003c/p\u003e\n\u003cp\u003eUltimately, a single, simple advice from the Prophet (PBUH) served as the foundation for a complete spiritual and moral transformation.\u003c/p\u003e", "language": "en" }, { "title" : "Migrating from Blogger to Hugo", "date_published" : "2026-06-21T00:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/migrasi-dari-blogger-ke-hugo/", "url" : "https://www.aantriono.com/en/migrasi-dari-blogger-ke-hugo/", "summary": "This article walks through the process of migrating from Blogger to Hugo, from exporting data and restructuring content to the final publishing step.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article walks through the process of migrating from Blogger to Hugo, from exporting data and restructuring content to the final publishing step.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"why-move-from-blogger-to-hugo\" class=\"headerLink\"\u003e\n \u003ca href=\"#why-move-from-blogger-to-hugo\" class=\"header-mark\" aria-label=\"Direct link to section: Why Move from Blogger to Hugo?\"\u003e\u003c/a\u003eWhy Move from Blogger to Hugo?\u003c/h2\u003e\u003cp\u003eBlogger is practical when you want to start a blog quickly. Publishing is simple, and the setup overhead is minimal. But as the number of posts grows, the requirements usually change as well. You may want a cleaner content structure, better performance, more control over the theme, and a more reliable backup workflow.\nAt that point, Hugo becomes a strong option because:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eHugo generates fast static sites.\u003c/li\u003e\n\u003cli\u003eAll content is stored in files, so backups are straightforward.\u003c/li\u003e\n\u003cli\u003eFolder structure and templates are much more flexible.\u003c/li\u003e\n\u003cli\u003eLocal development is easier to manage.\u003c/li\u003e\n\u003cli\u003ePublishing can be integrated with Git and static hosting platforms.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"what-to-prepare\" class=\"headerLink\"\u003e\n \u003ca href=\"#what-to-prepare\" class=\"header-mark\" aria-label=\"Direct link to section: What to Prepare\"\u003e\u003c/a\u003eWhat to Prepare\u003c/h2\u003e\u003cp\u003eBefore starting the migration, prepare the following:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eA Blogger export file in XML format.\u003c/li\u003e\n\u003cli\u003eHugo installed on your local machine.\u003c/li\u003e\n\u003cli\u003eA Hugo project repository.\u003c/li\u003e\n\u003cli\u003eA list of posts, labels, and important pages you want to keep.\u003c/li\u003e\n\u003cli\u003eA URL plan if you want to preserve already indexed links.\nIf the old blog already gets traffic from search engines, the goal is not only to move the content but also to preserve important metadata and URL structure to reduce SEO impact.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"1-export-content-from-blogger\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-export-content-from-blogger\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Export Content from Blogger\"\u003e\u003c/a\u003e1. Export Content from Blogger\u003c/h2\u003e\u003cp\u003eOpen the Blogger dashboard, then go to:\n\u003ccode\u003eSettings -\u0026gt; Manage blog -\u0026gt; Back up content -\u0026gt; Download\u003c/code\u003e\nBlogger will generate a single XML file containing posts, pages, comments, labels, and other metadata. Keep this file safely because it becomes the main source for the migration.\nIt is also a good idea to create additional backups:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ecopy the list of static pages,\u003c/li\u003e\n\u003cli\u003edocument important widgets,\u003c/li\u003e\n\u003cli\u003esave the old Blogger theme if you still need it as a reference,\u003c/li\u003e\n\u003cli\u003enote the most visited article URLs.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"2-create-or-prepare-a-hugo-project\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-create-or-prepare-a-hugo-project\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Create or Prepare a Hugo Project\"\u003e\u003c/a\u003e2. Create or Prepare a Hugo Project\u003c/h2\u003e\u003cp\u003eIf you do not already have a Hugo project, create one first:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ehugo new site blog-baru\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen choose and install a theme, and start the local server:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ehugo server\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eGet familiar with the main Hugo structure early:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003econtent/\u003c/code\u003e for posts and pages,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003elayouts/\u003c/code\u003e for templates,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003estatic/\u003c/code\u003e for static assets,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eassets/\u003c/code\u003e for the resource pipeline,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003econfig/\u003c/code\u003e or \u003ccode\u003ehugo.toml\u003c/code\u003e for site configuration.\nIf you already have an active Hugo project, this stage mainly means making sure the content structure is ready to receive the migrated posts.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"3-convert-blogger-posts-to-hugo-format\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-convert-blogger-posts-to-hugo-format\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Convert Blogger Posts to Hugo Format\"\u003e\u003c/a\u003e3. Convert Blogger Posts to Hugo Format\u003c/h2\u003e\u003cp\u003eThe Blogger XML file cannot be used directly as Hugo content. You need to convert it into Markdown files, usually in a structure like:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003econtent/posts/post-name/index.md\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eor\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003econtent/posts/post-name.md\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eA minimal Hugo content file usually contains front matter and the article body:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003eyaml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nn\"\u003e---\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003etitle\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Article Title\u0026#34;\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003edate\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"ld\"\u003e2026-06-21\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003edraft\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"kc\"\u003efalse\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003etags\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Blogger\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Hugo\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003ecategories\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Tutorials\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nn\"\u003e---\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe parts that usually need to be mapped from Blogger to Hugo are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003epost title,\u003c/li\u003e\n\u003cli\u003epublication date,\u003c/li\u003e\n\u003cli\u003elabels converted into \u003ccode\u003etags\u003c/code\u003e,\u003c/li\u003e\n\u003cli\u003eold permalink,\u003c/li\u003e\n\u003cli\u003eHTML or Markdown body content,\u003c/li\u003e\n\u003cli\u003eimages embedded in the article.\nIf you use your own migration script, make sure the output is consistent and does not mix multiple front matter styles.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"4-clean-up-the-front-matter\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-clean-up-the-front-matter\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Clean Up the Front Matter\"\u003e\u003c/a\u003e4. Clean Up the Front Matter\u003c/h2\u003e\u003cp\u003eAfter conversion, do not assume every post is ready to publish immediately. Migrated front matter usually still needs cleanup.\nCommon Hugo fields include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003etitle\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003edate\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003elastmod\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003edraft\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003edescription\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003etags\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003ecategories\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003ealiases\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eurl\u003c/code\u003e\nExample:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003eyaml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nn\"\u003e---\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003etitle\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Migrating from Blogger to Hugo\u0026#34;\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003edate\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"ld\"\u003e2026-06-21\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003elastmod\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"ld\"\u003e2026-06-21\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003edraft\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"kc\"\u003efalse\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003edescription\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;A tutorial on migrating content from Blogger to Hugo.\u0026#34;\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003etags\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Hugo\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Blogger\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Tutorial\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003ecategories\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;Tutorials\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nt\"\u003ealiases\u003c/span\u003e\u003cspan class=\"p\"\u003e:\u003c/span\u003e\u003cspan class=\"w\"\u003e \u003c/span\u003e\u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s2\"\u003e\u0026#34;/2023/07/migrasi-dari-blogger-ke-hugo.html\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e]\u003c/span\u003e\u003cspan class=\"w\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"w\"\u003e\u003c/span\u003e\u003cspan class=\"nn\"\u003e---\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe \u003ccode\u003ealiases\u003c/code\u003e field is especially useful if you want visitors opening an old URL to be redirected to the new one.\u003c/p\u003e\n\u003ch2 id=\"5-handle-old-urls\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-handle-old-urls\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Handle Old URLs\"\u003e\u003c/a\u003e5. Handle Old URLs\u003c/h2\u003e\u003cp\u003eThis is one of the most commonly overlooked parts. In Blogger, article URLs usually look like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e/2023/07/article-title.html\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eMeanwhile, Hugo often uses a shorter slug, for example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e/article-title/\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you change every URL without a redirect strategy, search traffic and old backlinks can drop.\nSome approaches you can use:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eadd \u003ccode\u003ealiases\u003c/code\u003e to individual posts,\u003c/li\u003e\n\u003cli\u003epreserve the old structure with \u003ccode\u003eurl\u003c/code\u003e,\u003c/li\u003e\n\u003cli\u003econfigure redirects at the hosting layer,\u003c/li\u003e\n\u003cli\u003eprioritize redirects for the most visited articles.\nFor an established blog, preserving permalinks matters more than simply making the URLs look cleaner.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"6-review-images-and-media\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-review-images-and-media\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Review Images and Media\"\u003e\u003c/a\u003e6. Review Images and Media\u003c/h2\u003e\u003cp\u003eContent migration almost always leaves work to do on images. Many Blogger posts use image links from Blogger\u0026rsquo;s CDN or still contain old HTML attributes.\nCheck the following:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ewhether the images are still accessible,\u003c/li\u003e\n\u003cli\u003ewhether the image URLs are still valid,\u003c/li\u003e\n\u003cli\u003ewhether the image files are too large,\u003c/li\u003e\n\u003cli\u003ewhether the \u003ccode\u003ealt\u003c/code\u003e attributes are filled in,\u003c/li\u003e\n\u003cli\u003ewhether the images should be copied into the local Hugo project structure.\nThe safest strategy is usually:\u003c/li\u003e\n\u003cli\u003edownload important images,\u003c/li\u003e\n\u003cli\u003eplace them in each post folder or in \u003ccode\u003estatic/images/\u003c/code\u003e,\u003c/li\u003e\n\u003cli\u003eupdate the references inside the content.\nThat way, you are not overly dependent on external image hosts for older content.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"7-clean-up-blogger-html\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-clean-up-blogger-html\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Clean Up Blogger HTML\"\u003e\u003c/a\u003e7. Clean Up Blogger HTML\u003c/h2\u003e\u003cp\u003eBlogger often produces noisy HTML, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003enested \u003ccode\u003e\u0026lt;span\u0026gt;\u003c/code\u003e tags,\u003c/li\u003e\n\u003cli\u003elong inline styles,\u003c/li\u003e\n\u003cli\u003e`` tags used for simple formatting,\u003c/li\u003e\n\u003cli\u003epresentational attributes that are no longer needed,\u003c/li\u003e\n\u003cli\u003emarkup copied from the visual editor.\nIf you leave all of it untouched, the article may still render, but it becomes harder to maintain. After conversion:\u003c/li\u003e\n\u003cli\u003esimplify paragraphs,\u003c/li\u003e\n\u003cli\u003econvert what can become plain Markdown,\u003c/li\u003e\n\u003cli\u003eclean up headings,\u003c/li\u003e\n\u003cli\u003ereview ordered and unordered lists,\u003c/li\u003e\n\u003cli\u003eremove unnecessary inline styles.\nIf an older article is genuinely complex, it is still acceptable to keep parts of it as HTML as long as the rendering remains correct. The main goal is consistency and maintainability.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"8-migrate-important-pages-besides-posts\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-migrate-important-pages-besides-posts\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Migrate Important Pages Besides Posts\"\u003e\u003c/a\u003e8. Migrate Important Pages Besides Posts\u003c/h2\u003e\u003cp\u003eDo not focus only on articles. A Blogger site often also has pages such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eAbout,\u003c/li\u003e\n\u003cli\u003eContact,\u003c/li\u003e\n\u003cli\u003eTable of contents,\u003c/li\u003e\n\u003cli\u003eDisclaimer,\u003c/li\u003e\n\u003cli\u003ePrivacy policy,\u003c/li\u003e\n\u003cli\u003especial pages for downloads or profiles.\nThose pages should also be moved into Hugo, usually under \u003ccode\u003econtent/\u003c/code\u003e according to the site structure. Make sure the navigation menu in the Hugo theme points to the new pages correctly.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"9-test-the-migration-locally\" class=\"headerLink\"\u003e\n \u003ca href=\"#9-test-the-migration-locally\" class=\"header-mark\" aria-label=\"Direct link to section: 9. Test the Migration Locally\"\u003e\u003c/a\u003e9. Test the Migration Locally\u003c/h2\u003e\u003cp\u003eBefore deploying, run Hugo locally and inspect the result carefully.\nThings to check:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ewhether all posts appear,\u003c/li\u003e\n\u003cli\u003ewhether publication dates are correct,\u003c/li\u003e\n\u003cli\u003ewhether tags and categories are recognized,\u003c/li\u003e\n\u003cli\u003ewhether internal links still work,\u003c/li\u003e\n\u003cli\u003ewhether images display properly,\u003c/li\u003e\n\u003cli\u003ewhether the table of contents, shortcodes, or code blocks remain clean,\u003c/li\u003e\n\u003cli\u003ewhether older posts with custom URLs are redirected correctly.\nThis step takes time, but it is the part that prevents bigger issues after the site goes live.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"10-deploy-to-hosting\" class=\"headerLink\"\u003e\n \u003ca href=\"#10-deploy-to-hosting\" class=\"header-mark\" aria-label=\"Direct link to section: 10. Deploy to Hosting\"\u003e\u003c/a\u003e10. Deploy to Hosting\u003c/h2\u003e\u003cp\u003eOnce the local result is clean, you can build the site:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ebash\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ehugo\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output is usually generated in the \u003ccode\u003epublic/\u003c/code\u003e folder. From there, the site can be published to platforms such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eGitHub Pages,\u003c/li\u003e\n\u003cli\u003eCloudflare Pages,\u003c/li\u003e\n\u003cli\u003eNetlify,\u003c/li\u003e\n\u003cli\u003eVercel,\u003c/li\u003e\n\u003cli\u003ea private server.\nIf your workflow is Git-based, deployment becomes much simpler and easier to repeat.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"common-challenges\" class=\"headerLink\"\u003e\n \u003ca href=\"#common-challenges\" class=\"header-mark\" aria-label=\"Direct link to section: Common Challenges\"\u003e\u003c/a\u003eCommon Challenges\u003c/h2\u003e\u003cp\u003eSome common issues during migration from Blogger to Hugo are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eOld article HTML is too messy.\u003c/li\u003e\n\u003cli\u003eImages still depend on the old host.\u003c/li\u003e\n\u003cli\u003eOld URLs are not preserved.\u003c/li\u003e\n\u003cli\u003eBlogger labels are inconsistent.\u003c/li\u003e\n\u003cli\u003eArticle metadata is incomplete.\u003c/li\u003e\n\u003cli\u003eThe Hugo theme does not yet support all display needs.\nBecause of that, migration should also be treated as a content cleanup process, not just a file transfer.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"a-safer-migration-strategy\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-safer-migration-strategy\" class=\"header-mark\" aria-label=\"Direct link to section: A Safer Migration Strategy\"\u003e\u003c/a\u003eA Safer Migration Strategy\u003c/h2\u003e\u003cp\u003eIf you have many articles, do not migrate everything at once without verification. A safer approach is:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003emigrate a subset of posts first,\u003c/li\u003e\n\u003cli\u003ereview the result,\u003c/li\u003e\n\u003cli\u003erefine the conversion script,\u003c/li\u003e\n\u003cli\u003econtinue with the next batch,\u003c/li\u003e\n\u003cli\u003erun a final audit before full publication.\nA staged approach usually reduces rework.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eMigrating from Blogger to Hugo gives you more control over content, performance, and publishing workflow. But a good result does not come from export and import alone. The important work is in cleaning up content structure, mapping old URLs, and verifying the final output.\nIf the migration is done carefully, Hugo can become a much stronger long-term foundation for managing a blog.\u003c/p\u003e", "language": "en" }, { "title" : "Lines and Angles", "date_published" : "2025-12-15T22:14:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/garis-dan-sudut/", "url" : "https://www.aantriono.com/en/garis-dan-sudut/", "summary": " Understanding AnglesAn angle is formed by two rays that meet at the same endpoint. Angles are denoted by the symbol ∠\\angle∠. In an angle, we recognize the angle arms, the vertex, and the angle region.\nIn the picture, the arms of the angle are rays OAOAOA and OBOBOB. Point OOO is the vertex, while the shaded area is the angle region.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"understanding-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#understanding-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Understanding Angles\"\u003e\u003c/a\u003eUnderstanding Angles\u003c/h2\u003e\u003cp\u003eAn angle is formed by two rays that meet at the same endpoint. Angles are denoted by the symbol \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. In an angle, we recognize the angle arms, the vertex, and the angle region.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRxHObz7aHc90e-RL_X83z8W77VBzGnWKGMxsvTrG9CiqIT1HV2hz_FkK_FUFn_WqK6HwI0sm6AU8Lm6r8U-gcFfoCdaJs4h2Mn-ZNulAZ-aeaneVzFuAT5_IjmSR029esXjm7SNriGJSx3tRmtJoEBcSYmnxkv0pHtcjt5DMQ9q8X6BhOidGRB521/s1280/sudut%20(1).png' alt=\"Grade 7 lines and angles material\" \u003e\u003c/p\u003e\n\u003cp\u003eIn the picture, the arms of the angle are rays \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eOA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eOB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eOB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eO\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the vertex, while the shaded area is the angle region.\u003c/p\u003e\n\u003cp\u003eNotes:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe vertex is the intersection point of the two angle arms;\u003c/li\u003e\n\u003cli\u003ethe angle region is the area bounded by the two arms.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"angle-measures\" class=\"headerLink\"\u003e\n \u003ca href=\"#angle-measures\" class=\"header-mark\" aria-label=\"Direct link to section: Angle Measures\"\u003e\u003c/a\u003eAngle Measures\u003c/h2\u003e\u003cp\u003eDegrees are the most common unit for measuring angles. One full turn is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. If a full turn is divided into 360 equal parts, each part measures \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eSmaller units than degrees are minutes and seconds:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmsup\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmsup\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3600\u003c/mn\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n1^\\circ=60\u0026#x27;, \\qquad 1\u0026#x27;=60\u0026#x27;\u0026#x27;, \\qquad 1^\\circ=3600\u0026#x27;\u0026#x27;\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9963em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9963em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e360\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"conversion-example\" class=\"headerLink\"\u003e\n \u003ca href=\"#conversion-example\" class=\"header-mark\" aria-label=\"Direct link to section: Conversion Example\"\u003e\u003c/a\u003eConversion Example\u003c/h3\u003e\u003cp\u003eFind the equivalent values:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmo\u003e…\u003c/mo\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmo\u003e…\u003c/mo\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^\\circ= \\ldots \u0026#x27;= \\ldots \u0026#x27;\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e7200\u003c/mn\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmo\u003e…\u003c/mo\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmo\u003e…\u003c/mo\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7200\u0026#x27;\u0026#x27;= \\ldots \u0026#x27;= \\ldots ^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e720\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eAnswer:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3^\\circ=(3\\times 60)\u0026#x27;=180\u0026#x27;\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0519em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3600\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10800\u003c/mn\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3^\\circ=(3\\times 3600)\u0026#x27;\u0026#x27;=10800\u0026#x27;\u0026#x27;\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0519em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3600\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1080\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e7200\u003c/mn\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmfrac\u003e\u003cmn\u003e7200\u003c/mn\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e120\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n7200\u0026#x27;\u0026#x27;=\\frac{7200}{60}\u0026#x27;=120\u0026#x27;\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e720\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1492em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7200\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4632em;\"\u003e\u003cspan style=\"top:-3.7743em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e7200\u003c/mn\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003cmo mathvariant=\"normal\"\u003e′\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmfrac\u003e\u003cmn\u003e7200\u003c/mn\u003e\u003cmn\u003e3600\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n7200\u0026#x27;\u0026#x27;=\\frac{7200}{3600}^\\circ=2^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e720\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0715em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3600\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7200\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3855em;\"\u003e\u003cspan style=\"top:-3.7743em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"addition-and-subtraction-of-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#addition-and-subtraction-of-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Addition and Subtraction of Angles\"\u003e\u003c/a\u003eAddition and Subtraction of Angles\u003c/h2\u003e\u003cp\u003eFind the results:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e12^\\circ 25\u0026#x27; + 13^\\circ 11\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8352em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e35\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e35^\\circ 11\u0026#x27; - 21^\\circ 10\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8352em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eAnswer:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n12^\\circ 25\u0026#x27; + 13^\\circ 11\u0026#x27; = 25^\\circ 36\u0026#x27;\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8852em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e35\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e01\u003c/mn\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n35^\\circ 11\u0026#x27; - 21^\\circ 10\u0026#x27; = 14^\\circ 01\u0026#x27;\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8852em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"drawing-and-naming-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-and-naming-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing and Naming Angles\"\u003e\u003c/a\u003eDrawing and Naming Angles\u003c/h2\u003e\u003ch3 id=\"drawing-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Angles\"\u003e\u003c/a\u003eDrawing Angles\u003c/h3\u003e\u003cp\u003eTo draw an angle with a known measure, we can use a ruler and a protractor.\u003c/p\u003e\n\u003cp\u003eA protractor has two scales:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eif the angle is measured clockwise, use the outer scale;\u003c/li\u003e\n\u003cli\u003eif the angle is measured counterclockwise, use the inner scale.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExample: draw an angle of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e70\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e70^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eSteps:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eDraw a straight line \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003ePlace the protractor on line \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with its center exactly at point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRfKkPeZHM9KMiUowmTbNJo3Qt4aEGB5pkM59Mp4ZQp1fDB9zuS_PiJCQ8lFMXfD9BuTD1oUQ8v57q3ssiyTZnI-D9-6KH-jWEtkSDZK-YYvkQ2GInpzx4Ig3P6UowJa98jbsIkheTLlA1nfP1NMv72iUhu7-vbJQ7Fq9rLNR7Jv44p9q2R2Ur2GWb/s592/busur1.png' alt=\"Drawing an angle with a protractor\" \u003e\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eMark point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e70\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e70^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eConnect points \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to form the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e70\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e70^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e angle.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_Rk2pz-B_au_73MzF6q60pXpi1OAyB2O37YiUKtEIPlu0SFAK6SzpPZ9gZlzb-4aZj2l_BH86wa7OqHckU-RRLiwpcbvJOKG6YJwc-fep2bads0gANCA9q4AcBikkTLcGOmGHl9VWyDNYxe4NdMVHG3nw7yyLT59wmFqH_GQQuofQW8qshxCvnJFU/s568/busur2.png' alt=\"Measuring an angle with a protractor\" \u003e\u003c/p\u003e\n\u003ch3 id=\"angle-notation-and-names\" class=\"headerLink\"\u003e\n \u003ca href=\"#angle-notation-and-names\" class=\"header-mark\" aria-label=\"Direct link to section: Angle Notation and Names\"\u003e\u003c/a\u003eAngle Notation and Names\u003c/h3\u003e\u003cp\u003eAngles are usually named in three ways:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eusing symbols such as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\beta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\theta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eθ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e;\u003c/li\u003e\n\u003cli\u003eusing the name of the vertex;\u003c/li\u003e\n\u003cli\u003eusing three capital letters, with the vertex letter in the middle.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg18l7x0Y9CDFVZ1Of1Tvw4PnVWwVUNTRqW7Q9ckCEOrvmu_a90mCnMHIOa4JOLBVLtR_VoIfAjr0mdL5UyBxhvSvp3O-RSGrrnCN7BqE-7Rpg_nEm59dcJEsB5jXsCf-890V0n5RFGF_pkZ_j-5HA9UQMyuKYhIkGrwbnfcoMdPGlcaVGoKqZUbqOV/s360/Screenshot%202022-04-09%2009.01.21.png' alt=\"Naming an angle\" \u003e\u003c/p\u003e\n\u003cp\u003eThe angle in the figure may be named \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eα\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle \\alpha\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle A\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle BAC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle CAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch2 id=\"types-of-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Types of Angles\"\u003e\u003c/a\u003eTypes of Angles\u003c/h2\u003e\u003cp\u003eBased on their measure, angles are divided into:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eacute angles, less than \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e;\u003c/li\u003e\n\u003cli\u003eright angles, exactly \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e;\u003c/li\u003e\n\u003cli\u003eobtuse angles, more than \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and less than \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e;\u003c/li\u003e\n\u003cli\u003estraight angles, exactly \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e;\u003c/li\u003e\n\u003cli\u003ereflex angles, more than \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and less than \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJrTEkF1MkGSfrMoRqj0fM6xd2rjFLv8AieLssDsl1ZOH28U5sjPTbZQG_Na6NcEdbvUGHT1Af8-dG4B7yc4IaJGUPJU8V4yLdAxwJnap0_HTQeVdwJ6yGXlm4gSiXUjqWFMvGXjwh-5BLuMCi6Xi8YKYnjxECU6EIlw9I6IVddseaLz3JiLaP9VTs/s537/Screenshot%202022-04-11%2021.18.48.png' alt=\"Types of angles\" \u003e\u003c/p\u003e\n\u003ch2 id=\"relationships-between-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#relationships-between-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Relationships Between Angles\"\u003e\u003c/a\u003eRelationships Between Angles\u003c/h2\u003e\u003ch3 id=\"complementary-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#complementary-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Complementary Angles\"\u003e\u003c/a\u003eComplementary Angles\u003c/h3\u003e\u003cp\u003eThe sum of two complementary angles is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYUmoa0KJ9jgPD5eL5QG5sblknnnxYs_XmJcMRZcIUy1L_A5U6bMCq_EufBmlP6lblVI72p52Dbo0aWkjHF20C6h9qE1itQSl-xKLHQjK4ga10cqDUsx8iadVXgVZXYy-iryxyeEg7GvyN34QqpZmFRHmZD2kYQipwk4hmGWQQTxtGMJQ5ToMaxYCL/s303/Screenshot%202022-04-11%2021.36.24.png' alt=\"Complementary angles\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle AOC+\\angle BOC=\\angle AOB\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7756em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eOC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBOC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eOB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^\\circ+y^\\circ=90^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8074em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9186em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"supplementary-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#supplementary-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Supplementary Angles\"\u003e\u003c/a\u003eSupplementary Angles\u003c/h3\u003e\u003cp\u003eThe sum of two supplementary angles is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlideU60oVoRo9aFn351ZmVTtctGvIX4DNEDcJ1sr30qWZ2SZ-41o2QZ2LOcPDUx4VH55FAD3KkCiH-WhSwY4b1XZT8zShmCF6-o05rZRq_Lv40p01I1pM3e5CeDcPUTqjH7hWfpTlpWyltk8e58RmIwi7EHGLstvur2TiSiUfuHs3iNuaCt9ETfnJ/s303/Screenshot%202022-04-11%2021.36.55.png' alt=\"Supplementary angles\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle AOC+\\angle BOC=\\angle AOB\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7756em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eOC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBOC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eOB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^\\circ+y^\\circ=180^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8074em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9186em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"vertical-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#vertical-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Vertical Angles\"\u003e\u003c/a\u003eVertical Angles\u003c/h3\u003e\u003cp\u003eIf two lines intersect, the opposite angles are called vertical angles. Their measures are equal.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXK5401WeehW3l_tn_MVbqHZ2qfPsYKlikQSgXYDVngY_d3DKtm6Y0rpJbhfcycLIPsiro3InJyqEgvPSqMnlooAhGyePpZU2lZR2txcc_PQxcBDWfZfe3ii5ucEga_C3xfMMbILhWCKRbhPYGzHMpmV2bOudmFE7jqXTN8SHJJBQFNtK9yUW3V6VT/s322/Screenshot%202022-04-11%2021.42.02.png' alt=\"Vertical angles\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle AOB=\\angle COD\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eOB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eCO\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle AOD=\\angle BOC\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBOC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-problems\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problems\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problems\"\u003e\u003c/a\u003eExample Problems\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eObserve the following figure.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZnDGVPZQimWYk1o5V1FfsyO1SEldcJJcSNM065Tvff0A8Qdr2Chp_Vh929VVr1PmkZNKNTMq2XJ92-qrtWKp7DrsmmqgZn1X7F79qxNh0-ORkw9kfbPNnlk_wkV8aVCSCVTmhGz7xVJ4kyS97gry1Nb1CD28-bcx1talXksxqvhiwBY5xPdkfiiSx/s316/Screenshot%202022-04-13%2008.44.10.png' alt=\"Complementary angle example\" \u003e\u003c/p\u003e\n\u003cp\u003eFind:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e;\u003c/li\u003e\n\u003cli\u003ethe measure of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle POR\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003ePOR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eAnswer:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2x+3x=90^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e18\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n5x=90^\\circ \\Rightarrow x=18^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle POR=2x=36^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003ePOR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eObserve the following figure.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0apXug_g8WZ8qfd1jiL0W8YmVyOLt17plMcvpobbBWOmKVVn7Pj4RnlL1NqF-Ii6Cv-kIEsqPXgOBwH7eZpv27tKvHJu4ROATlOpbB-yMMyd1DrKK_sTpp4UwdxXy-62TT9qy3Qyd9Sjw-lCB6rzKDsmaN89nvq3dus5wmQDAPl4TL0s9KTWuCHGu/s316/Screenshot%202022-04-13%2008.47.44.png' alt=\"Supplementary angle example\" \u003e\u003c/p\u003e\n\u003cp\u003eThe supplementary angle of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle KLN\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3x+15+2x+10=180\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n5x+25=180\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e155\u003c/mn\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e31\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n5x=155 \\Rightarrow x=31\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e155\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e31\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e72\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle KLN=\\angle MLN=(2x+10)^\\circ=72^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"3\"\u003e\n\u003cli\u003eIn the following figure, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle LOM=100^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eOM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Find \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle KOL\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle KON\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eON\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDYxiBmVoX0FJWTj25DGfdOoOSti1F7hzOBANpupJ00pj8q9au5VKlXQzQOokOoC_8Na2Xd2_zS0cuXeriheKVXBkCoGfocPD2QdSHKNzrtdM64BUYkxq3CiLozIhJhyCy33UwxEwVY1nwuVALcT1bwCIV9XVbtkUDKWvhwlNU3M06hSeJpu6bdn-z/s338/Screenshot%202022-06-04%2009.17.37.png' alt=\"Supplementary and vertical angle example\" \u003e\u003c/p\u003e\n\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle KOL\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle LOM\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eOM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are supplementary:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle KOL+100^\\circ=180^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7756em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e80\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle KOL=80^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle KON\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eON\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle LOM\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eOM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are vertical angles:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle KON=100^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eON\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#lines\" class=\"header-mark\" aria-label=\"Direct link to section: Lines\"\u003e\u003c/a\u003eLines\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw8CtRgntedDRD_BW8cRqQSY7g014fLtmIOrelVoTQQ2_e-oPw_Ysz2HFRfen-bV2TJrk89Wl9xIKIYbnOW0z4Uwu9eEuw4GowSl3XzTdlY8DSSk5AhkcLCf7OEjsYbSElmYvwh-xBYRWRDUFZsoUs-cWRKXmGeTuaV5lz26oZygD3N4VRobkDSEpF/s629/Screenshot%202022-04-13%2009.01.22.png' alt=\"Line segment, ray, and line\" \u003e\u003c/p\u003e\n\u003cp\u003eFigure (a) is a line segment \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, written \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. If end \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is extended, we get ray \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, written \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e→\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overrightarrow{AB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2053em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord accent\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2053em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"svg-align\" style=\"top:-3.6833em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"height:0.522em;min-width:0.888em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"0.522em\" viewBox=\"0 0 400000 522\" preserveAspectRatio=\"xMaxYMin slice\"\u003e\u003cpath d=\"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. If both ends are extended indefinitely, we get line \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, written \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e↔\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overleftrightarrow{AB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2053em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord accent\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2053em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"svg-align\" style=\"top:-3.6833em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"stretchy\" style=\"height:0.522em;min-width:0.888em;\"\u003e\u003cspan class=\"halfarrow-left\" style=\"height:0.522em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"0.522em\" viewBox=\"0 0 400000 522\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M400000 241H110l3-3c68.7-52.7 113.7-120\n 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8\n-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247\nc-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208\n 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3\n 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202\n l-3-3h399890zM100 241v40h399900v-40z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003cspan class=\"halfarrow-right\" style=\"height:0.522em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"0.522em\" viewBox=\"0 0 400000 522\" preserveAspectRatio=\"xMaxYMin slice\"\u003e\u003cpath d=\"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch2 id=\"position-of-two-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#position-of-two-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Position of Two Lines\"\u003e\u003c/a\u003ePosition of Two Lines\u003c/h2\u003e\u003cp\u003eThere are four possible positions of two lines: parallel, intersecting, coinciding, and skew.\u003c/p\u003e\n\u003ch3 id=\"parallel-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#parallel-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Parallel Lines\"\u003e\u003c/a\u003eParallel Lines\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNh7nmO0btgdTKZ9LU5tjdT57PIWwwjT1c42kaxWTVqFxIf11mvXYE2BsQwkMcyzdkeYPVbtuMJ6VD0kU_RjO72E7e_goESWSXESs43h-WUDukEc5UhoN-zUBCNYHOawtszWLuBmP3uaKaqhvZtlp9b006hl3_qW4s2hVtARhgKTkbTq73dZG2xa8k/s286/Screenshot%202022-05-10%2021.26.54.png' alt=\"Parallel lines\" \u003e\u003c/p\u003e\n\u003cp\u003eLine \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is parallel to line \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, written \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e∥\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\parallel b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eNote: two or more lines are parallel if they lie in the same plane and never intersect even when extended.\u003c/p\u003e\n\u003ch3 id=\"intersecting-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#intersecting-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Intersecting Lines\"\u003e\u003c/a\u003eIntersecting Lines\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1cQsir-2XEiPiluU28NcHW8UYEaV3B9WEAWLqJ0PZnGPbK7S_UxGXLwTcFy-3Xw3kPIw2La_zcbgxWB4BgYSlJZ19p84eQE6gITcGsFGmYAMk7Cv00WxvfuKRn63JuwUkdzsudGwtQRRI36dUjf570UHYkiqjyh0l4r05cp-603A0JlgIbV2Ljtgd/s286/Screenshot%202022-05-10%2021.27.09.png' alt=\"Intersecting lines\" \u003e\u003c/p\u003e\n\u003cp\u003eTwo lines intersect if they have exactly one common point.\u003c/p\u003e\n\u003ch3 id=\"coinciding-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#coinciding-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Coinciding Lines\"\u003e\u003c/a\u003eCoinciding Lines\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZI9TVD2c7yu3pK4jJANoSxMsoTeF7iWLbyF8s1PEFw-tv8Nwo6XwiIbJL_-F7MHRY_itYz9f_EzoRiuFL62O2kWF4xl1m8hAJ1zLMEaUNz0PhhbuPYc-bec3RFwpZE3etceTOXmEjpaOoTz8oHD2bMZoF1gb6RTnA1n2aIPUTXut17SRULqen4FwK/s264/Screenshot%202022-05-10%2021.27.27.png' alt=\"Coinciding lines\" \u003e\u003c/p\u003e\n\u003cp\u003eTwo lines coincide if they lie on the same straight line and appear as a single line.\u003c/p\u003e\n\u003ch3 id=\"skew-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#skew-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Skew Lines\"\u003e\u003c/a\u003eSkew Lines\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimRvciZ-OGKoR5PAudTRe32XrQ-55zV5zjFrk8JOAgI-SYGXitYNjiPjHWnzl9jKITi6HrtWm25KrFWwXiCJ-okUIX5eyWZMGUiLGGqIKEWRlCcSXpH0jc9JBF_Lmj9vnv0V0jJt55c9izSWcAhelgABd4fCNmQMG0DuU5me43LJ5SzU_iCDNzvEDx/s403/Screenshot%202022-05-10%2021.28.05.png' alt=\"Skew lines\" \u003e\u003c/p\u003e\n\u003cp\u003eSkew lines only appear in three-dimensional figures. They do not lie on the same plane and do not intersect even when extended.\u003c/p\u003e\n\u003ch2 id=\"properties-of-parallel-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#properties-of-parallel-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Properties of Parallel Lines\"\u003e\u003c/a\u003eProperties of Parallel Lines\u003c/h2\u003e\u003cp\u003eAn axiom is a statement accepted as true without proof.\u003c/p\u003e\n\u003cp\u003eOne example of an axiom:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethrough a point outside a line, exactly one line can be drawn parallel to that line.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eSome properties of parallel lines:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eIf a line intersects one of two parallel lines, it also intersects the other.\u003c/li\u003e\n\u003cli\u003eIf a line is parallel to two other lines, then those two lines are parallel to each other.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"relationship-between-lines-and-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#relationship-between-lines-and-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Relationship Between Lines and Angles\"\u003e\u003c/a\u003eRelationship Between Lines and Angles\u003c/h2\u003e\u003cp\u003eIn the figure below, line \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e intersects two parallel lines, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003el\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003el\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.01968em;\"\u003el\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Line \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called a transversal.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlx626Uw_YVnAlUmN18ALg7RWpAWtGuTGAdevSpc8yCunflLDgeaajfK2XqzxocJ-iP-HQOidLlNeJ99JYwcNz0dRMOHSP4GwjFSRg1vBP9_95mRRq6xHLre-t7_M7Ecog4VfUpu7gm4slLb6PpE1iNRjas0xkJo7-8Rz4QUSbA8SB7-d0TgTAes9K/s433/Screenshot%202022-06-02%2010.53.23.png' alt=\"Two parallel lines cut by another line\" \u003e\u003c/p\u003e\n\u003cp\u003eAt point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e there are four angles: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. At point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e there are also four angles: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle Q_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle Q_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle Q_3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle Q_4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"corresponding-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#corresponding-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Corresponding Angles\"\u003e\u003c/a\u003eCorresponding Angles\u003c/h3\u003e\u003cp\u003eEach pair of corresponding angles has the same measure:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_1=\\angle Q_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_2=\\angle Q_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_3=\\angle Q_3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_4=\\angle Q_4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"alternate-interior-and-alternate-exterior-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#alternate-interior-and-alternate-exterior-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Alternate Interior and Alternate Exterior Angles\"\u003e\u003c/a\u003eAlternate Interior and Alternate Exterior Angles\u003c/h3\u003e\u003cp\u003eEach pair of alternate interior angles is equal:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_3=\\angle Q_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_4=\\angle Q_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eEach pair of alternate exterior angles is also equal:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_1=\\angle Q_3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_2=\\angle Q_4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cstrong\u003eGeoGebra simulation\u003c/strong\u003e\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/pwhznvkf/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/false/ctl/false\" style=\"border: 0px;\" title=\"Sudut Berseberangan\" width=\"700px\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"same-side-interior-and-same-side-exterior-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#same-side-interior-and-same-side-exterior-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Same-Side Interior and Same-Side Exterior Angles\"\u003e\u003c/a\u003eSame-Side Interior and Same-Side Exterior Angles\u003c/h3\u003e\u003cp\u003eEach pair of same-side interior angles sums to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_3+\\angle Q_2=180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_4+\\angle Q_1=180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eEach pair of same-side exterior angles also sums to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_1+\\angle Q_4=180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle P_2+\\angle Q_3=180^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8422em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cstrong\u003eGeoGebra simulation\u003c/strong\u003e\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/uepmnjtm/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/false/ctl/false\" style=\"border: 0px;\" title=\"Sudut Sepihak\" width=\"700px\"\u003e \u003c/iframe\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat concludes the material about lines and angles. If you notice any mistake or want to add a suggestion, please leave a comment.\u003c/p\u003e\n", "language": "en" }, { "title" : "How to Study Effectively and Efficiently", "date_published" : "2025-12-15T21:58:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-belajar-efektif-dan-efisien/", "url" : "https://www.aantriono.com/en/cara-belajar-efektif-dan-efisien/", "summary": " How to Study Effectively and EfficientlyAre people smart from birth? Of course not. People need to learn in order to become smart. Through learning, we can discover things that were previously unknown. Still, sometimes people study many times and still do not understand what they have learned. Why does that happen?\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"how-to-study-effectively-and-efficiently\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-study-effectively-and-efficiently\" class=\"header-mark\" aria-label=\"Direct link to section: How to Study Effectively and Efficiently\"\u003e\u003c/a\u003eHow to Study Effectively and Efficiently\u003c/h2\u003e\u003cp\u003eAre people smart from birth? Of course not. People need to learn in order to become smart. Through learning, we can discover things that were previously unknown. Still, sometimes people study many times and still do not understand what they have learned. Why does that happen?\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOrzc2DAax_pZl2n1V4_C-a_RVJwAOVedEdr9g-A0wOmvhzM8KH9rDRIcVEtrmTAVwzTbRsgu46LhZaXnEpiB6Mt0pWKSqEXSBMtOsW0T4dOzp4UogIi0R1jX197kL6iYruWyyBNn_EDaNX5e2g12CFDoO6t7dpMFndevkZqVl69q6XtMChvT-lmGP/w640-h200/baca%20buku.png\" title=\"How to study effectively\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOrzc2DAax_pZl2n1V4_C-a_RVJwAOVedEdr9g-A0wOmvhzM8KH9rDRIcVEtrmTAVwzTbRsgu46LhZaXnEpiB6Mt0pWKSqEXSBMtOsW0T4dOzp4UogIi0R1jX197kL6iYruWyyBNn_EDaNX5e2g12CFDoO6t7dpMFndevkZqVl69q6XtMChvT-lmGP/w640-h200/baca%20buku.png\" data-sub-html=\"\u003ch2\u003eHow to study effectively and efficiently\u003c/h2\u003e\u003cp\u003eHow to study effectively\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOrzc2DAax_pZl2n1V4_C-a_RVJwAOVedEdr9g-A0wOmvhzM8KH9rDRIcVEtrmTAVwzTbRsgu46LhZaXnEpiB6Mt0pWKSqEXSBMtOsW0T4dOzp4UogIi0R1jX197kL6iYruWyyBNn_EDaNX5e2g12CFDoO6t7dpMFndevkZqVl69q6XtMChvT-lmGP/w640-h200/baca%20buku.png' alt=\"How to study effectively\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eHow to study effectively and efficiently\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eOne reason may be that the study method is not effective and efficient. In fact, anyone can study. The difference is that people who understand how to study well usually absorb the material better. So how do you study more effectively and efficiently?\u003c/p\u003e\n\u003cp\u003eHere are a few things that can help.\u003c/p\u003e\n\u003ch3 id=\"straighten-your-intention\" class=\"headerLink\"\u003e\n \u003ca href=\"#straighten-your-intention\" class=\"header-mark\" aria-label=\"Direct link to section: Straighten Your Intention\"\u003e\u003c/a\u003eStraighten Your Intention\u003c/h3\u003e\u003cp\u003eStart with the right intention. Study to gain knowledge, understand it, and apply it in real life.\u003c/p\u003e\n\u003ch3 id=\"set-goals\" class=\"headerLink\"\u003e\n \u003ca href=\"#set-goals\" class=\"header-mark\" aria-label=\"Direct link to section: Set Goals\"\u003e\u003c/a\u003eSet Goals\u003c/h3\u003e\u003cp\u003eSet a clear study target. For example, aim for a certain score on an exam or work toward admission to a specific university. Clear goals help you stay focused and consistent.\u003c/p\u003e\n\u003ch3 id=\"set-a-study-schedule\" class=\"headerLink\"\u003e\n \u003ca href=\"#set-a-study-schedule\" class=\"header-mark\" aria-label=\"Direct link to section: Set a Study Schedule\"\u003e\u003c/a\u003eSet a Study Schedule\u003c/h3\u003e\u003cp\u003eDiscipline matters. For example, study for one hour each evening and take a short break in between. Regular study is better than last-minute cramming.\u003c/p\u003e\n\u003ch3 id=\"create-a-comfortable-study-environment\" class=\"headerLink\"\u003e\n \u003ca href=\"#create-a-comfortable-study-environment\" class=\"header-mark\" aria-label=\"Direct link to section: Create a Comfortable Study Environment\"\u003e\u003c/a\u003eCreate a Comfortable Study Environment\u003c/h3\u003e\u003cp\u003eChoose a study environment that helps you stay focused. Some people work best in silence, while others prefer a little music. Know yourself and choose what works best.\u003c/p\u003e\n\u003ch3 id=\"make-a-summary\" class=\"headerLink\"\u003e\n \u003ca href=\"#make-a-summary\" class=\"header-mark\" aria-label=\"Direct link to section: Make a Summary\"\u003e\u003c/a\u003eMake a Summary\u003c/h3\u003e\u003cp\u003eWrite down the key points from what you study. Summaries help you understand the core material and make review easier before exams.\u003c/p\u003e\n\u003ch3 id=\"use-easy-to-remember-acronyms\" class=\"headerLink\"\u003e\n \u003ca href=\"#use-easy-to-remember-acronyms\" class=\"header-mark\" aria-label=\"Direct link to section: Use Easy-to-Remember Acronyms\"\u003e\u003c/a\u003eUse Easy-to-Remember Acronyms\u003c/h3\u003e\u003cp\u003eIf you need to remember sequences or terms, use acronyms. This can make memorization much easier.\u003c/p\u003e\n\u003ch3 id=\"understand-not-memorize\" class=\"headerLink\"\u003e\n \u003ca href=\"#understand-not-memorize\" class=\"header-mark\" aria-label=\"Direct link to section: Understand, Not Memorize\"\u003e\u003c/a\u003eUnderstand, Not Memorize\u003c/h3\u003e\u003cp\u003eMemorizing alone usually helps only in the short term. If you understand the material, it stays with you longer and is easier to apply.\u003c/p\u003e\n\u003ch3 id=\"do-not-be-afraid-to-ask\" class=\"headerLink\"\u003e\n \u003ca href=\"#do-not-be-afraid-to-ask\" class=\"header-mark\" aria-label=\"Direct link to section: Do Not Be Afraid to Ask\"\u003e\u003c/a\u003eDo Not Be Afraid to Ask\u003c/h3\u003e\u003cp\u003eIf something is unclear, ask questions. You can ask friends, teachers, or discuss it with others. Asking often speeds up understanding.\u003c/p\u003e\n\u003ch3 id=\"never-give-up\" class=\"headerLink\"\u003e\n \u003ca href=\"#never-give-up\" class=\"header-mark\" aria-label=\"Direct link to section: Never Give Up\"\u003e\u003c/a\u003eNever Give Up\u003c/h3\u003e\u003cp\u003eIf the results are still not where you want them to be, that does not mean you have failed. Stay calm, review your method, and try again.\u003c/p\u003e\n\u003cp\u003eThese are some ways to study effectively and efficiently. Keep learning.\u003c/p\u003e\n", "language": "en" }, { "title" : "Eid Greeting Card", "date_published" : "2025-12-14T13:32:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kartu-ucapan-lebaran/", "url" : "https://www.aantriono.com/en/kartu-ucapan-lebaran/", "summary": " Making Eid Greeting Cards with LaTeXQ Unconsciously, for Muslims we have entered the final week of the fasting month of Ramadan 1442 H. When it comes to the end of Ramadhan, usually fellow Muslims will send each other Eid greeting cards via social media such as Instagram, Facebook, Twitter and also via messaging applications such as WhatsApp, Telegram, Line, and so on. Currently, there are lots of various kinds of Eid greeting cards circulating, we can choose and download them for free on the internet, of course by including the credit title on the Eid greeting card.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"making-eid-greeting-cards-with-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#making-eid-greeting-cards-with-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Making Eid Greeting Cards with LaTeX\"\u003e\u003c/a\u003eMaking Eid Greeting Cards with LaTeX\u003c/h2\u003e\u003cp\u003eQ\nUnconsciously, for Muslims we have entered the final week of the fasting month of Ramadan 1442 H. When it comes to the end of Ramadhan, usually fellow Muslims will send each other Eid greeting cards via social media such as Instagram, Facebook, Twitter and also via messaging applications such as WhatsApp, Telegram, Line, and so on. Currently, there are lots of various kinds of Eid greeting cards circulating, we can choose and download them for free on the internet, of course by including the credit title on the Eid greeting card.\u003c/p\u003e\n\u003ch3 id=\"making-eid-greeting-cards-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#making-eid-greeting-cards-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Making Eid Greeting Cards in LaTeX\"\u003e\u003c/a\u003eMaking Eid Greeting Cards in LaTeX\u003c/h3\u003e\u003cp\u003eWe can design our own Eid greeting cards according to what we want. In LaTeX, we can use the pgfornament package to frame Eid greetings so that they appear more elegant. Readers certainly still remember how \u003ca href=\"https://www.silagebra.com/2021/04/menghias-dokumen-dengan-ornamen.html\" target=\"_blank\"\u003edecorate documents with ornaments\u003c/a\u003e\nwhich I previously wrote about on this blog. We just have to change the ornaments and word choices according to what we want. OK, let\u0026rsquo;s just discuss how to make Eid greeting cards in LaTeX.\nThe first step is to determine the class of documents and packages needed. In this case I use the document class \u003cem\u003escartcl.\u003c/em\u003e\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[paper=a4,landscape]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escrartcl\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\areaset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e0.9\u003cspan class=\"k\"\u003e\\paperwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.9\u003cspan class=\"k\"\u003e\\paperheight\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pagestyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eempty\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eNext, to make the writing more beautiful, we will use the Calligra font, by first activating the T1 Fontenc package in the preamble.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[T1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efontenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalligra\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen we use the pgfornament package, and the TikZ library {calc} to determine the coordinates, via the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfornament\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"manufacturing-steps\" class=\"headerLink\"\u003e\n \u003ca href=\"#manufacturing-steps\" class=\"header-mark\" aria-label=\"Direct link to section: Manufacturing Steps\"\u003e\u003c/a\u003eManufacturing Steps\u003c/h3\u003e\u003cp\u003eNow in the body of the document we write the commands.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eevery node/.style = \u003cspan class=\"nb\"\u003e{\u003c/span\u003einner sep = 0pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003epgfornamentstyle/.style = \u003cspan class=\"nb\"\u003e{\u003c/span\u003ecolor = green!50!black,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003efill = green!80!black\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [text width = 11cm, outer sep = 1.7cm,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003etext centered, color = red!90!black] (Greeting)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen, for each corner of the Eid greeting card, an ornament with the code {63} will be installed using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\corner\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\sym\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003enorth west/none, north east/v,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esouth west/h, south east/c\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e [anchor = \u003cspan class=\"k\"\u003e\\corner\u003c/span\u003e] (\u003cspan class=\"k\"\u003e\\corner\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eat (Greeting.\u003cspan class=\"k\"\u003e\\corner\u003c/span\u003e)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width = 2.75cm, symmetry = \\sym]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e63\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eNext, for the middle right and left of the card, we will give ornaments with the code {9} using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e (north west) -- (south west)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003enode [midway, anchor = east]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[height = 2cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e9\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e(north east) -- (south east)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003enode [midway, anchor = west]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e[height = 2cm,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003esymmetry = v]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e9\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eNext, for the top part, an ornament will be given with the code {87} via the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfornamenthline\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003enorth west\u003cspan class=\"nb\"\u003e}{\u003c/span\u003enorth east\u003cspan class=\"nb\"\u003e}{\u003c/span\u003enorth\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e87\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd the bottom part is also given the same ornament as the top part, namely an ornament with the code {87} via the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfornamenthline\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esouth west\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esouth east\u003cspan class=\"nb\"\u003e}{\u003c/span\u003esouth\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e87\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd of course in the end we will write greetings for Eid al-Fitr 1442 H, using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\Large\u003c/span\u003e \u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\calligra\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003eAan Triono dan keluarga mengucapkan :\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSelamat Hari Raya Idul Fitri 1442 H\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eTaqabbalallahu minna wa minkum,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eshiyamanaa wa shiyamakum, kullu \u0026#39;aamin wa antum bi\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ekhoir, minal \u0026#39;aidin wal faizin\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"na\"\u003e[1ex]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eOf course, you can change and adapt to your needs any ornaments applied, coloring or greetings given. And finally we will add an ornament below the greeting given via the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e[color = red!90!black,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ewidth = 3.5cm]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e83\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd don\u0026rsquo;t forget to close the script by writing the command:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"the-final-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-final-result\" class=\"header-mark\" aria-label=\"Direct link to section: The final result\"\u003e\u003c/a\u003eThe final result\u003c/h3\u003e\u003cp\u003eThe final result of the manuscript that has been written using the steps above looks like in the following image.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwXlqFvdv3A_XHRX4R2jlzUlHMb0lLEEmK4ZLYh1HaWOyRyuwTNAXx8dJfR4lj0_qfca_AghH-Ow0hZ236c74DFjjUZkFlR826ncufaOWXNWkvBEbz150fNBctX0f96wLrrgze246jssQ/s900/kartulebaran.png\"\u003e\n\u003cimg alt=\"kartu ucapan lebaran di latex\" border=\"0\" data-original-height=\"504\" data-original-width=\"900\" height=\"179\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwXlqFvdv3A_XHRX4R2jlzUlHMb0lLEEmK4ZLYh1HaWOyRyuwTNAXx8dJfR4lj0_qfca_AghH-Ow0hZ236c74DFjjUZkFlR826ncufaOWXNWkvBEbz150fNBctX0f96wLrrgze246jssQ/w320-h179/kartulebaran.png\" title=\"kartu ucapan lebaran\" width=\"320\" /\u003e\n\u003c/a\u003e\nThat\u0026rsquo;s a guide on how to make Eid greeting cards in LaTeX. If there are errors, please correct them, and if you have input, please write them here. I hope this is helpful.\u003c/p\u003e", "language": "en" }, { "title" : "Set Intersections", "date_published" : "2025-12-14T13:29:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/irisan-himpunan/", "url" : "https://www.aantriono.com/en/irisan-himpunan/", "summary": " Venn Diagram: Set IntersectionsThe intersection of sets A and B is the set containing elements that belong to both A and B. It is written as:\nA∩B={x∣x∈A and x∈B} A \\cap B = \\{x \\mid x \\in A \\text{ and } x \\in B\\} A∩B={x∣x∈A and x∈B}In a Venn diagram, the universal set is usually shown as a rectangle, while the other sets are represented by circles or ellipses. The overlapping region represents the intersection.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"venn-diagram-set-intersections\" class=\"headerLink\"\u003e\n \u003ca href=\"#venn-diagram-set-intersections\" class=\"header-mark\" aria-label=\"Direct link to section: Venn Diagram: Set Intersections\"\u003e\u003c/a\u003eVenn Diagram: Set Intersections\u003c/h2\u003e\u003cp\u003eThe intersection of sets \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e is the set containing elements that belong to both \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e. It is written as:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∣\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmtext\u003e and \u003c/mtext\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA \\cap B = \\{x \\mid x \\in A \\text{ and } x \\in B\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∣\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7335em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIn a Venn diagram, the universal set is usually shown as a rectangle, while the other sets are represented by circles or ellipses. The overlapping region represents the intersection.\u003c/p\u003e\n\u003cp\u003eConsider the following example.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTLnDTpYAOlZPu0JYqNUnfCJ6gf9GzaXDR-wryZCsfzomQ3GZXfCwl73RHW2PKqR8PLJkbVygZsce1rcA_lnQ0Ey2BYqflrGtqenV_qJJQceu7URm7DzsVvZlcC5jyjgMb17mQyd3-17A/s589/Screenshot_5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"intersection of two sets\" border=\"0\" data-original-height=\"511\" data-original-width=\"589\" height=\"278\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTLnDTpYAOlZPu0JYqNUnfCJ6gf9GzaXDR-wryZCsfzomQ3GZXfCwl73RHW2PKqR8PLJkbVygZsce1rcA_lnQ0Ey2BYqflrGtqenV_qJJQceu7URm7DzsVvZlcC5jyjgMb17mQyd3-17A/w320-h278/Screenshot_5.png\" title=\"set intersection\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"example-problem\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003eS = {1,2,3,4,5,6,7,8,9,10,11,12,13}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eA = {the first six natural numbers}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eB = {the first six prime numbers}\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eDetermine:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eThe members of set \u003ccode\u003eA\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eThe members of set \u003ccode\u003eB\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eThe members of \u003ccode\u003eA \\cap B\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eThe Venn diagram for the problem\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"drawing-the-intersection-of-two-sets-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-the-intersection-of-two-sets-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing the Intersection of Two Sets in LaTeX\"\u003e\u003c/a\u003eDrawing the Intersection of Two Sets in LaTeX\u003c/h3\u003e\u003cp\u003eTo draw a diagram like the example above, you can combine \u003ccode\u003etikz\u003c/code\u003e and \u003ccode\u003etcolorbox\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eRequired packages:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eintersections,patterns\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[many]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tcbuselibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eskins\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eExample frame setup:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tcbset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e frogbox/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e enhanced,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colback=white!10,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colframe=green!65!black,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e enlarge top by=5.5mm,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e overlay=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2cm,3.5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e([xshift=\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e]frame.north west)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=green!65!black,fill=gray!20,line width=1mm]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (0,0) arc (0:180:5mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=black]\u003c/span\u003e (-0.2,0) arc (0:180:1mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the document body, start with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[frogbox,title=\u003cspan class=\"k\"\u003e\\large\u003c/span\u003e Venn Diagram]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe frame looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrxEu7s3EUtZ983gzcFSfGRCjfnzWfNqw6QwVPgmrJN2FMOH_QO-lnZPcs1kQu3k46o_9agy_pHDlliEntuRTBu4Q1Vo4hYD1dJwaE8gesev1H5H7kM_4NVgTB1OygwChcc0a274SBhMI/s1323/Screenshot_3.png\"\u003e\u003cimg alt=\"frog-eye frame in latex\" border=\"0\" data-original-height=\"138\" data-original-width=\"1323\" height=\"66\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrxEu7s3EUtZ983gzcFSfGRCjfnzWfNqw6QwVPgmrJN2FMOH_QO-lnZPcs1kQu3k46o_9agy_pHDlliEntuRTBu4Q1Vo4hYD1dJwaE8gesev1H5H7kM_4NVgTB1OygwChcc0a274SBhMI/w640-h66/Screenshot_3.png\" title=\"frog-eye frame\" width=\"640\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch4 id=\"creating-the-circles-and-guide-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-the-circles-and-guide-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Creating the Circles and Guide Lines\"\u003e\u003c/a\u003eCreating the Circles and Guide Lines\u003c/h4\u003e\u003cp\u003eStart the drawing area with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[fill opacity=.4]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTo make positioning easier, draw the universal-set rectangle and guide lines:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (-5,5) rectangle (5,-3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[help lines]\u003c/span\u003e (-5,5) grid (5,-3);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen draw the two intersecting circles:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=A,draw,fill=green]\u003c/span\u003e (-1.5,1) circle (3) node(NA)\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=B,draw,fill=blue]\u003c/span\u003e (1.5,1) circle (3) node(NB)\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe temporary result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkTRIALlZPD_6nJ-bbJw-IjULS_Doob9RJ-RQtU6FP8EBStYjYSFXMDDGbZJ5nxuoA8dExaui24mqa488F4n6dNgdBBBDZLJj49ryuzkv2wVSkrZnh56iurS8PhgOx1TvvTiwQxAJ8kS0/s1262/Screenshot_4.png\"\u003e\u003cimg alt=\"drawing the intersection of two sets\" border=\"0\" data-original-height=\"491\" data-original-width=\"1262\" height=\"125\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkTRIALlZPD_6nJ-bbJw-IjULS_Doob9RJ-RQtU6FP8EBStYjYSFXMDDGbZJ5nxuoA8dExaui24mqa488F4n6dNgdBBBDZLJj49ryuzkv2wVSkrZnh56iurS8PhgOx1TvvTiwQxAJ8kS0/w320-h125/Screenshot_4.png\" title=\"intersection of two sets\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch4 id=\"highlighting-the-intersection-region\" class=\"headerLink\"\u003e\n \u003ca href=\"#highlighting-the-intersection-region\" class=\"header-mark\" aria-label=\"Direct link to section: Highlighting the Intersection Region\"\u003e\u003c/a\u003eHighlighting the Intersection Region\u003c/h4\u003e\u003cp\u003eTo emphasize the overlap, use:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[fill opacity=1]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e (NB) circle (2.99297);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (NA) circle (2.99297);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen label the sets:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (-4.8,4.8) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eS\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (-3,4) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eA\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (3,4) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eB\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch4 id=\"inserting-the-set-members\" class=\"headerLink\"\u003e\n \u003ca href=\"#inserting-the-set-members\" class=\"header-mark\" aria-label=\"Direct link to section: Inserting the Set Members\"\u003e\u003c/a\u003eInserting the Set Members\u003c/h4\u003e\u003cp\u003eMembers of set \u003ccode\u003eA\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (-3,2) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (-2,1) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e4\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (-3,0) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e6\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eMembers of set \u003ccode\u003eB\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ered\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e7\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (2,1) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ered\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e11\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (3,0) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ered\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e13\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan 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401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 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8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (0,2) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (0,1) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e3\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (0,0) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e5\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eMembers of the universal set \u003ccode\u003eS\u003c/code\u003e outside \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (-4,-2) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e8\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (-4,-2.5) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e9\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (4,-2) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e10\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e at (4,-2.5) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\cdot\u003c/span\u003e\u003cspan class=\"m\"\u003e12\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eOnce everything is placed correctly, remove the guide lines if they are no longer needed. The final result will match the first image.\u003c/p\u003e\n\u003ch3 id=\"drawing-the-intersection-of-three-sets-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-the-intersection-of-three-sets-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing the Intersection of Three Sets in LaTeX\"\u003e\u003c/a\u003eDrawing the Intersection of Three Sets in LaTeX\u003c/h3\u003e\u003cp\u003eTo draw the intersection of three sets, add a third circle, say set \u003ccode\u003eC\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[name path=C,draw,fill=red]\u003c/span\u003e (0,-2) circle (3) node(NC)\u003cspan class=\"nb\"\u003e{}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you no longer want the \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eB\u003c/code\u003e overlap to be filled white as before, replace the command with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[]\u003c/span\u003e (NA) circle (0);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen add the members of set \u003ccode\u003eC\u003c/code\u003e as needed. The result may look like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQJPOuAHdja71lX9R74BDb3XehtLlcSZ1HxQOKxcPtkag_Uhg08GIZ7QME1XNLA4jEtFnNSxvHeVjm1VeGRrpXM5n1tWuxx0rne8nrN5COy8yCHPQymuOTyP53_hyphenhyphencb_CY8yxHP-pRjJg/s501/Screenshot_6.png\"\u003e\u003cimg alt=\"venn diagram of three-circle intersection\" border=\"0\" data-original-height=\"501\" data-original-width=\"492\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQJPOuAHdja71lX9R74BDb3XehtLlcSZ1HxQOKxcPtkag_Uhg08GIZ7QME1XNLA4jEtFnNSxvHeVjm1VeGRrpXM5n1tWuxx0rne8nrN5COy8yCHPQymuOTyP53_hyphenhyphencb_CY8yxHP-pRjJg/w314-h320/Screenshot_6.png\" title=\"intersection of three circles\" width=\"314\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eWith TikZ, Venn diagrams for two-set and three-set intersections can be drawn quite flexibly. This approach works well for worksheets, teaching material, and practice problems that need a clean visual representation of set operations.\u003c/p\u003e\n", "language": "en" }, { "title" : "Setting Page Margins", "date_published" : "2025-12-14T13:24:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mengatur-margin-halaman/", "url" : "https://www.aantriono.com/en/mengatur-margin-halaman/", "summary": " Setting Page Margins in LaTeXWhen writing a document in LaTeX\\LaTeXLATE​X, page margins should be defined early so the printed result matches the intended layout. A page margin is the empty space between the document content and the edge of the paper.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"setting-page-margins-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#setting-page-margins-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Setting Page Margins in LaTeX\"\u003e\u003c/a\u003eSetting Page Margins in LaTeX\u003c/h2\u003e\u003cp\u003eWhen writing a document in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, page margins should be defined early so the printed result matches the intended layout. A page margin is the empty space between the document content and the edge of the paper.\u003c/p\u003e\n\u003cp\u003eDocument classes such as \u003ccode\u003earticle\u003c/code\u003e already provide default margin settings. If you want to inspect that layout, you can use the \u003ccode\u003elayout\u003c/code\u003e package.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003elayout\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\layout\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe compiled result will show the page layout information.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG51xH00P8rDCcBfvbC14in7vUuhDeOPPDsVWoWDmfmmlFsS8S6ct653uNvzd4VxDKZ7MBhWQTTJ2DiO0ifgGq5PhAvISehYPoB5w4FSBOLpEUPlIUQYtPFgI9BfHcFQc67rrYMDFSC7g/w226-h320/0001\u0026#43;%252817%2529.jpg\" title=\"Page layout information\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG51xH00P8rDCcBfvbC14in7vUuhDeOPPDsVWoWDmfmmlFsS8S6ct653uNvzd4VxDKZ7MBhWQTTJ2DiO0ifgGq5PhAvISehYPoB5w4FSBOLpEUPlIUQYtPFgI9BfHcFQc67rrYMDFSC7g/w226-h320/0001\u0026#43;%252817%2529.jpg\" data-sub-html=\"\u003ch2\u003ePage layout information\u003c/h2\u003e\u003cp\u003ePage layout information\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG51xH00P8rDCcBfvbC14in7vUuhDeOPPDsVWoWDmfmmlFsS8S6ct653uNvzd4VxDKZ7MBhWQTTJ2DiO0ifgGq5PhAvISehYPoB5w4FSBOLpEUPlIUQYtPFgI9BfHcFQc67rrYMDFSC7g/w226-h320/0001+%252817%2529.jpg' alt=\"Page layout information\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003ePage layout information\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG51xH00P8rDCcBfvbC14in7vUuhDeOPPDsVWoWDmfmmlFsS8S6ct653uNvzd4VxDKZ7MBhWQTTJ2DiO0ifgGq5PhAvISehYPoB5w4FSBOLpEUPlIUQYtPFgI9BfHcFQc67rrYMDFSC7g/s2048/0001\u0026#43;%252817%2529.jpg\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG51xH00P8rDCcBfvbC14in7vUuhDeOPPDsVWoWDmfmmlFsS8S6ct653uNvzd4VxDKZ7MBhWQTTJ2DiO0ifgGq5PhAvISehYPoB5w4FSBOLpEUPlIUQYtPFgI9BfHcFQc67rrYMDFSC7g/s2048/0001+%252817%2529.jpg\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eTo set the top, left, bottom, and right margins directly, the most practical package is \u003ccode\u003egeometry\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eBasic format:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[top=distance,left=distance,bottom=distance,right=distance]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"positive-margins\" class=\"headerLink\"\u003e\n \u003ca href=\"#positive-margins\" class=\"header-mark\" aria-label=\"Direct link to section: Positive Margins\"\u003e\u003c/a\u003ePositive Margins\u003c/h3\u003e\u003cp\u003eThe following example sets the top margin to 3 cm, left 3 cm, bottom 3 cm, and right 2.5 cm.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[a4paper,12pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[top=3cm,left=3cm,bottom=3cm,right=2.5cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eI am setting page margins in LaTeX.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRDDPdxDlirVDLOytIeGeGbuaBnbUBAwiPhobv6Nwra0CJMuD1KVp4ATULdzK0sANEuLr275Rp4lm6EHnpEc9_aG10u4xd20A_ZS7j5upm4EKVEVjvmATNCmlAJAuy4_3I3qLrueFZneo/w320-h126/margin1.png\" title=\"Positive margin example\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRDDPdxDlirVDLOytIeGeGbuaBnbUBAwiPhobv6Nwra0CJMuD1KVp4ATULdzK0sANEuLr275Rp4lm6EHnpEc9_aG10u4xd20A_ZS7j5upm4EKVEVjvmATNCmlAJAuy4_3I3qLrueFZneo/w320-h126/margin1.png\" data-sub-html=\"\u003ch2\u003ePositive margin example\u003c/h2\u003e\u003cp\u003ePositive margin example\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRDDPdxDlirVDLOytIeGeGbuaBnbUBAwiPhobv6Nwra0CJMuD1KVp4ATULdzK0sANEuLr275Rp4lm6EHnpEc9_aG10u4xd20A_ZS7j5upm4EKVEVjvmATNCmlAJAuy4_3I3qLrueFZneo/w320-h126/margin1.png' alt=\"Positive margin example\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003ePositive margin example\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRDDPdxDlirVDLOytIeGeGbuaBnbUBAwiPhobv6Nwra0CJMuD1KVp4ATULdzK0sANEuLr275Rp4lm6EHnpEc9_aG10u4xd20A_ZS7j5upm4EKVEVjvmATNCmlAJAuy4_3I3qLrueFZneo/s1249/margin1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRDDPdxDlirVDLOytIeGeGbuaBnbUBAwiPhobv6Nwra0CJMuD1KVp4ATULdzK0sANEuLr275Rp4lm6EHnpEc9_aG10u4xd20A_ZS7j5upm4EKVEVjvmATNCmlAJAuy4_3I3qLrueFZneo/s1249/margin1.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eYou do not have to specify every side. For example, this sets only the top and left margins:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[a4paper,12pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[top=3cm,left=3cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eI am setting page margins in LaTeX.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf all margins should be the same, use the shorthand \u003ccode\u003emargin\u003c/code\u003e option.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[a4paper,12pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=3cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"zero-and-negative-margins\" class=\"headerLink\"\u003e\n \u003ca href=\"#zero-and-negative-margins\" class=\"header-mark\" aria-label=\"Direct link to section: Zero and Negative Margins\"\u003e\u003c/a\u003eZero and Negative Margins\u003c/h3\u003e\u003cp\u003eThe \u003ccode\u003egeometry\u003c/code\u003e package also allows a margin of \u003ccode\u003e0cm\u003c/code\u003e or even a negative value.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[top=0cm,left=3cm,bottom=3cm,right=-3cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003elipsum\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\lipsum\u003c/span\u003e\u003cspan class=\"na\"\u003e[1-3]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eWith that setup, the text can start exactly at the top edge and even extend past the right boundary.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL0yFi8nJZ_OTz5pc9vppGWTtLboB-xiJn_yfnQSdodNUJ_aUbSpu8t9C1tpMknWxbcoUY6lLBe0R7bAIB-0AAyz9A12FsNARIsCoRfRn5p_cR_HgiatMgk8ZD_eQYe473ysycbRa_XfA/w400-h95/margin2.png\" title=\"Zero and negative margin example\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL0yFi8nJZ_OTz5pc9vppGWTtLboB-xiJn_yfnQSdodNUJ_aUbSpu8t9C1tpMknWxbcoUY6lLBe0R7bAIB-0AAyz9A12FsNARIsCoRfRn5p_cR_HgiatMgk8ZD_eQYe473ysycbRa_XfA/w400-h95/margin2.png\" data-sub-html=\"\u003ch2\u003eZero and negative margin example\u003c/h2\u003e\u003cp\u003eZero and negative margin example\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL0yFi8nJZ_OTz5pc9vppGWTtLboB-xiJn_yfnQSdodNUJ_aUbSpu8t9C1tpMknWxbcoUY6lLBe0R7bAIB-0AAyz9A12FsNARIsCoRfRn5p_cR_HgiatMgk8ZD_eQYe473ysycbRa_XfA/w400-h95/margin2.png' alt=\"Zero and negative margin example\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eZero and negative margin example\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL0yFi8nJZ_OTz5pc9vppGWTtLboB-xiJn_yfnQSdodNUJ_aUbSpu8t9C1tpMknWxbcoUY6lLBe0R7bAIB-0AAyz9A12FsNARIsCoRfRn5p_cR_HgiatMgk8ZD_eQYe473ysycbRa_XfA/s1237/margin2.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL0yFi8nJZ_OTz5pc9vppGWTtLboB-xiJn_yfnQSdodNUJ_aUbSpu8t9C1tpMknWxbcoUY6lLBe0R7bAIB-0AAyz9A12FsNARIsCoRfRn5p_cR_HgiatMgk8ZD_eQYe473ysycbRa_XfA/s1237/margin2.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eThis kind of setup is usually reserved for special layouts, not ordinary documents.\u003c/p\u003e\n\u003ch3 id=\"different-margins-on-different-pages\" class=\"headerLink\"\u003e\n \u003ca href=\"#different-margins-on-different-pages\" class=\"header-mark\" aria-label=\"Direct link to section: Different Margins on Different Pages\"\u003e\u003c/a\u003eDifferent Margins on Different Pages\u003c/h3\u003e\u003cp\u003eYou can also change the margins in the middle of a document. For example, suppose page one uses 3 cm margins on all sides, but page two needs:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003etop: 4 cm\u003c/li\u003e\n\u003cli\u003eleft: 6 cm\u003c/li\u003e\n\u003cli\u003ebottom: 4 cm\u003c/li\u003e\n\u003cli\u003eright: 5 cm\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eUse \u003ccode\u003e\\newgeometry\u003c/code\u003e for that page.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newgeometry\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etop=4cm,left=6cm,bottom=4cm,right=5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEintxcMpA_EZX5WpxunUinfBhrfXae6LJ4V4yj3Mxk2BcAQGWtNKTW5wanwaJwixgejX0aYFrlPexl3MX5zBt5BYiRQmgugeEQ87cVmfwoDP90CdjRyJhGo9QFwyz643UXTWGjBo5qE6Wg/w320-h194/margin3.png\" title=\"Different margins on another page\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEintxcMpA_EZX5WpxunUinfBhrfXae6LJ4V4yj3Mxk2BcAQGWtNKTW5wanwaJwixgejX0aYFrlPexl3MX5zBt5BYiRQmgugeEQ87cVmfwoDP90CdjRyJhGo9QFwyz643UXTWGjBo5qE6Wg/w320-h194/margin3.png\" data-sub-html=\"\u003ch2\u003eDifferent margins on another page\u003c/h2\u003e\u003cp\u003eDifferent margins on another page\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEintxcMpA_EZX5WpxunUinfBhrfXae6LJ4V4yj3Mxk2BcAQGWtNKTW5wanwaJwixgejX0aYFrlPexl3MX5zBt5BYiRQmgugeEQ87cVmfwoDP90CdjRyJhGo9QFwyz643UXTWGjBo5qE6Wg/w320-h194/margin3.png' alt=\"Different margins on another page\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eDifferent margins on another page\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEintxcMpA_EZX5WpxunUinfBhrfXae6LJ4V4yj3Mxk2BcAQGWtNKTW5wanwaJwixgejX0aYFrlPexl3MX5zBt5BYiRQmgugeEQ87cVmfwoDP90CdjRyJhGo9QFwyz643UXTWGjBo5qE6Wg/s708/margin3.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEintxcMpA_EZX5WpxunUinfBhrfXae6LJ4V4yj3Mxk2BcAQGWtNKTW5wanwaJwixgejX0aYFrlPexl3MX5zBt5BYiRQmgugeEQ87cVmfwoDP90CdjRyJhGo9QFwyz643UXTWGjBo5qE6Wg/s708/margin3.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eTo return to the original margin settings from the \u003cem\u003epreamble\u003c/em\u003e, use:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\restoregeometry\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis command is usually placed at the beginning of the next page. If needed, add \u003ccode\u003e\\newpage\u003c/code\u003e to force a page break.\u003c/p\u003e\n\u003ch3 id=\"closing-note\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing-note\" class=\"header-mark\" aria-label=\"Direct link to section: Closing Note\"\u003e\u003c/a\u003eClosing Note\u003c/h3\u003e\u003cp\u003eIn most documents, page margins only need to be set once. But for special pages containing large images, tables, or full-page graphics, it is useful to temporarily redefine the margins and then restore them.\u003c/p\u003e\n\u003cp\u003eWith the \u003ccode\u003egeometry\u003c/code\u003e package, page margin control in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e becomes simple and flexible.\u003c/p\u003e\n", "language": "en" }, { "title" : "Decorate Documents With Ornaments", "date_published" : "2025-12-14T13:18:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menghias-dokumen-dengan-ornamen/", "url" : "https://www.aantriono.com/en/menghias-dokumen-dengan-ornamen/", "summary": " Decorating Documents with LaTeX OrnamentsInvitation cards, greeting cards, certificates, and title pages often look better when the text is framed with decorative ornaments. In LaTeX, this can be done with ornament packages such as pgfornament.\nHere is one example.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"decorating-documents-with-latex-ornaments\" class=\"headerLink\"\u003e\n \u003ca href=\"#decorating-documents-with-latex-ornaments\" class=\"header-mark\" aria-label=\"Direct link to section: Decorating Documents with LaTeX Ornaments\"\u003e\u003c/a\u003eDecorating Documents with LaTeX Ornaments\u003c/h2\u003e\u003cp\u003eInvitation cards, greeting cards, certificates, and title pages often look better when the text is framed with decorative ornaments. In LaTeX, this can be done with ornament packages such as \u003ccode\u003epgfornament\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eHere is one example.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_3W9yMatfny_3CcOo-048H2mlA1MgISiyFVLSUFTCQ6RwPADxxPEa3MbXGnVsworm3bd1L-O1HxpSwHTPwU6RcFlhzmHVxtkqhJy9mg5PaHuRvvQptLcIqwZaBLfCe2-JKUb27aUbraM/s1180/ornamen1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"latex ornament\" border=\"0\" data-original-height=\"481\" data-original-width=\"1180\" height=\"130\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_3W9yMatfny_3CcOo-048H2mlA1MgISiyFVLSUFTCQ6RwPADxxPEa3MbXGnVsworm3bd1L-O1HxpSwHTPwU6RcFlhzmHVxtkqhJy9mg5PaHuRvvQptLcIqwZaBLfCe2-JKUb27aUbraM/w320-h130/ornamen1.png\" title=\"latex ornament\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eIn this article we will build two styles:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eornaments only in the page corners\u003c/li\u003e\n\u003cli\u003eornaments around the full page border\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"ornaments-in-the-page-corners\" class=\"headerLink\"\u003e\n \u003ca href=\"#ornaments-in-the-page-corners\" class=\"header-mark\" aria-label=\"Direct link to section: Ornaments in the Page Corners\"\u003e\u003c/a\u003eOrnaments in the Page Corners\u003c/h3\u003e\u003cp\u003eIn the first example, ornaments are placed only in the four corners.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkE0sKItTCNZ03R3tZoOfBup_MibS-rhbsm_CJZaBuRdi0dHZR8L1i5CxE1pbCFOpgkEp9w4-vlCQI-jSg9PfBW0ciJJ8jyQnAvXYWUIFeolJoUmuopc_aJUgX5Mk3RLNJeEKidGs0hUI/s438/ornamen2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"latex ornament\" border=\"0\" data-original-height=\"433\" data-original-width=\"438\" height=\"316\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkE0sKItTCNZ03R3tZoOfBup_MibS-rhbsm_CJZaBuRdi0dHZR8L1i5CxE1pbCFOpgkEp9w4-vlCQI-jSg9PfBW0ciJJ8jyQnAvXYWUIFeolJoUmuopc_aJUgX5Mk3RLNJeEKidGs0hUI/w320-h316/ornamen2.png\" title=\"latex ornament\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003ePrepare the document class and required packages first.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[dvipsnames]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzrput\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfornament\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen start the drawing in the document body.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[color=Maroon,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eevery node/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003einner sep=0pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eHere the ornament color is \u003ccode\u003eMaroon\u003c/code\u003e. To make positioning easier, create a square guide and grid lines first.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[ultra thick]\u003c/span\u003e (-6,-6) rectangle (6,6);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[help lines]\u003c/span\u003e (-6,-6) grid (6,6);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe four corner ornaments can be placed like this:\u003c/p\u003e\n\u003cp\u003eTop right:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=north east]\u003c/span\u003e at (vecbox.north east)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=5cm,symmetry=v]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e61\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTop left:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=north west]\u003c/span\u003e at (vecbox.north west)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=5cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e61\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eBottom left:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=south west]\u003c/span\u003e at (vecbox.south west)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=5cm,symmetry=h]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e61\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eBottom right:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=south east]\u003c/span\u003e at (vecbox.south east)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=5cm,symmetry=c]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e61\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you want to place text inside the frame, use commands like these.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (0,1)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\huge\u003c/span\u003e Insert text here\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (0,0)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Large\u003c/span\u003e Or here\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe coordinates inside the parentheses control the position on the \u003ccode\u003ex\u003c/code\u003e and \u003ccode\u003ey\u003c/code\u003e axes. Once the layout is correct, the guide lines can be removed from the final version.\u003c/p\u003e\n\u003cp\u003eThe result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0qtxq0cst3njG6KrRY5CfsMYEiTwP4lLSjkZGCGkRNWzTIKst_M-R7fxUqjP6HA8pVKdMm_EjRblRjGqXxJI9q156FDW-TbouGOBAXcLl6-x0QP3lqY85wkTwtcU1SWDyBGcrXZqXpww/s1118/ornamen3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"latex ornament\" border=\"0\" data-original-height=\"493\" data-original-width=\"1118\" height=\"141\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0qtxq0cst3njG6KrRY5CfsMYEiTwP4lLSjkZGCGkRNWzTIKst_M-R7fxUqjP6HA8pVKdMm_EjRblRjGqXxJI9q156FDW-TbouGOBAXcLl6-x0QP3lqY85wkTwtcU1SWDyBGcrXZqXpww/w320-h141/ornamen3.png\" title=\"latex ornament\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"ornaments-around-the-full-page-border\" class=\"headerLink\"\u003e\n \u003ca href=\"#ornaments-around-the-full-page-border\" class=\"header-mark\" aria-label=\"Direct link to section: Ornaments Around the Full Page Border\"\u003e\u003c/a\u003eOrnaments Around the Full Page Border\u003c/h3\u003e\u003cp\u003eAfter placing ornaments in the corners, you can continue by filling the middle sections of the border as well.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOXWY0jnrd2k0iqbNfF2V59cLr49ZKV6gr4mKFSzdAsqVEWoYjzMFqgKM5A0CRCx4RQ3rqSpp9GIuj8GoFGq3A6vrY36XlCox5u53mzcNd6mRp6lg0ikeyQop1bv_q6I8OTSmVaPb-wUQ/s446/ornamen4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"latex border ornament\" border=\"0\" data-original-height=\"445\" data-original-width=\"446\" height=\"319\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOXWY0jnrd2k0iqbNfF2V59cLr49ZKV6gr4mKFSzdAsqVEWoYjzMFqgKM5A0CRCx4RQ3rqSpp9GIuj8GoFGq3A6vrY36XlCox5u53mzcNd6mRp6lg0ikeyQop1bv_q6I8OTSmVaPb-wUQ/w320-h319/ornamen4.png\" title=\"latex border ornament\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eIn this example:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe corner ornament uses code \u003ccode\u003e63\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003ethe middle border ornament uses code \u003ccode\u003e46\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eTop center:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=north]\u003c/span\u003e at (vecbox.north)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=7cm,symmetry=h]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e46\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eLeft center, rotated vertically:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=north,rotate=90]\u003c/span\u003e at (vecbox.west)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=7cm,symmetry=h]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e46\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eBottom center:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=south]\u003c/span\u003e at (vecbox.south)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=7cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e46\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eRight center:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[anchor=north,rotate=-90]\u003c/span\u003e at (vecbox.east)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=7cm,symmetry=h]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e46\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eYou can also add ornaments inside the page, not only along the border. For example, the following commands place a title and two animal ornaments:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (0,3.3)\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eRed\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Large\u003c/span\u003e A lion chases a goat\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (-2,1.7) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=3.5cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e158\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (3,1.7) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=2.5cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e104\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd these commands add plant ornaments:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (0,-1.5) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=2.5cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e10\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (2,-2.3) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=1cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e10\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rput\u003c/span\u003e (-2,-2.3) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfornament\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=1cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e10\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTheir positions can be adjusted simply by changing the coordinates.\u003c/p\u003e\n\u003cp\u003eThe final result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHMws07ufbZxXTqVmI-xgW9O8TwPbZbJa77LSd7CMC1jx7JbXfJCLygraaQ71SmHeEvOJdNgTiZe1f3RU17rB66nUT93ISJux3zdHlEeBd4bKCYkbG5-nLD4vdyb52YHIljnTTCQ7KdvA/s456/ornamen5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"453\" data-original-width=\"456\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHMws07ufbZxXTqVmI-xgW9O8TwPbZbJa77LSd7CMC1jx7JbXfJCLygraaQ71SmHeEvOJdNgTiZe1f3RU17rB66nUT93ISJux3zdHlEeBd4bKCYkbG5-nLD4vdyb52YHIljnTTCQ7KdvA/s320/ornamen5.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eBesides animal and plant ornaments, the \u003ccode\u003epgfornament\u003c/code\u003e package also includes many other decorative elements. You can browse them in the official CTAN documentation: \u003ca href=\"https://ctan.org/pkg/pgfornament\" target=\"_blank\"\u003eCTAN pgfornament\u003c/a\u003e.\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eWith \u003ccode\u003epgfornament\u003c/code\u003e, you can make LaTeX documents look much more decorative for greeting cards, invitations, certificates, or title pages. The main work is choosing the ornament code, size, and position carefully.\u003c/p\u003e\n", "language": "en" }, { "title" : "How to Draw Circle Elements", "date_published" : "2025-12-14T13:14:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-menggambar-unsur-unsur-lingkaran/", "url" : "https://www.aantriono.com/en/cara-menggambar-unsur-unsur-lingkaran/", "summary": " Drawing Circle Elements with LaTeXAs mentioned in the previous post, a circle is a simple closed curve that represents the locus of points at the same distance from a fixed point. That fixed point is called the center of the circle. Circle material is usually studied in grade VIII. At the beginning, students are introduced to the definition of a circle and its elements, as shown in the following figure. ", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-circle-elements-with-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-circle-elements-with-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Circle Elements with LaTeX\"\u003e\u003c/a\u003eDrawing Circle Elements with LaTeX\u003c/h2\u003e\u003cp\u003eAs mentioned in the previous post, a circle is a simple closed curve that represents the locus of points at the same distance from a fixed point. That fixed point is called the center of the circle. Circle material is usually studied in grade VIII. At the beginning, students are introduced to the definition of a circle and its elements, as shown in the following figure.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhY1jmDwLNjbiPhG-_eOo4mK-831P8K3x_IbDLQYyMiqqkK-kit90sUnHB5hc6XP2NZanaNex2XeZ9TvENFAY9Viimyb0mN71INgTPH-yPufpdg5y3FhrcMLaVi8Wo74iQ31m2oIEZr4EE/s502/unsurling1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n\u003cimg alt=\"menggambar unsur-unsur lingkaran\" border=\"0\" data-original-height=\"391\" data-original-width=\"502\" height=\"249\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhY1jmDwLNjbiPhG-_eOo4mK-831P8K3x_IbDLQYyMiqqkK-kit90sUnHB5hc6XP2NZanaNex2XeZ9TvENFAY9Viimyb0mN71INgTPH-yPufpdg5y3FhrcMLaVi8Wo74iQ31m2oIEZr4EE/w320-h249/unsurling1.png\" title=\"menggambar unsur-unsur lingkaran\" width=\"320\" /\u003e\n\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"circle-elements\" class=\"headerLink\"\u003e\n \u003ca href=\"#circle-elements\" class=\"header-mark\" aria-label=\"Direct link to section: Circle Elements\"\u003e\u003c/a\u003eCircle Elements\u003c/h3\u003e\u003cp\u003eAs shown in the figure, the circle is placed inside a framed box. To create that frame, you can refer to the article about \u003ca href=\"https://www.silagebra.com/2021/03/color-box.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ecolor box\u003c/a\u003e.\u003c/p\u003e\n\u003cp\u003eThe elements of the circle shown there are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ecenter,\u003c/li\u003e\n\u003cli\u003eradius,\u003c/li\u003e\n\u003cli\u003ediameter,\u003c/li\u003e\n\u003cli\u003esector,\u003c/li\u003e\n\u003cli\u003earc,\u003c/li\u003e\n\u003cli\u003echord,\u003c/li\u003e\n\u003cli\u003eapothem.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"guide-to-drawing-circle-elements\" class=\"headerLink\"\u003e\n \u003ca href=\"#guide-to-drawing-circle-elements\" class=\"header-mark\" aria-label=\"Direct link to section: Guide to Drawing Circle Elements\"\u003e\u003c/a\u003eGuide to Drawing Circle Elements\u003c/h3\u003e\u003cp\u003eThen how do you draw the circle elements as seen in the image in LaTeX? Check out the following guide.\u003c/p\u003e\n\u003ch3 id=\"1-preparation-of-initial-packages-and-orders\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-preparation-of-initial-packages-and-orders\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Preparation of Initial Packages and Orders\"\u003e\u003c/a\u003e1. Preparation of Initial Packages and Orders\u003c/h3\u003e\u003cp\u003eInclude the following packages and commands in the preamble to the document.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 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356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 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16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumitem\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[tikz,blur]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebclogo\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalc,patterns,angles,quotes\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[many]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e%--- Paket yang dibutuhkan untuk membuat color box ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath,amssymb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newtcolorbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emyfancybox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[1]\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebreakable,enhanced jigsaw,colback=blue!20,colframe=red,top=20pt,overlay unbroken= \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=blue,text=yellow]\u003c/span\u003e at (frame.east) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\clubsuit\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Bagian untuk mengubah pewarnaan color box ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003eattach boxed title to top left=\u003cspan class=\"nb\"\u003e{\u003c/span\u003exshift=75pt,yshift=-7pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e, \u003cspan class=\"c\"\u003e%--- Bagian untuk mengubah posisi judul ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003eboxed title style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003esize=small,colback=blue,colframe=red\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003etitle=#1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e%--- Bagian untuk mengubah pewarnaan judul ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newcommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\siku\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[4][.3cm]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (tempa) at (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003e#\u003c/span\u003e\u003cspan class=\"m\"\u003e3\u003c/span\u003e\u003cspan class=\"o\"\u003e)!\u003c/span\u003e\u003cspan class=\"nb\"\u003e#\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"o\"\u003e!(\u003c/span\u003e\u003cspan class=\"nb\"\u003e#\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (tempb) at (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003e#\u003c/span\u003e\u003cspan class=\"m\"\u003e3\u003c/span\u003e\u003cspan class=\"o\"\u003e)!\u003c/span\u003e\u003cspan class=\"nb\"\u003e#\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"o\"\u003e!(\u003c/span\u003e\u003cspan class=\"nb\"\u003e#\u003c/span\u003e\u003cspan class=\"m\"\u003e4\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e (tempc) at (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003etempa\u003c/span\u003e\u003cspan class=\"o\"\u003e)!\u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"nb\"\u003e.\u003c/span\u003e\u003cspan class=\"m\"\u003e5\u003c/span\u003e\u003cspan class=\"o\"\u003e!(\u003c/span\u003e\u003cspan class=\"nb\"\u003etempb\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e (tempa) -- (\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003e#\u003c/span\u003e\u003cspan class=\"m\"\u003e3\u003c/span\u003e\u003cspan class=\"o\"\u003e)!\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"o\"\u003e!(\u003c/span\u003e\u003cspan class=\"nb\"\u003etempc\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e) -- (tempb);\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\noindent\u003c/span\u003e Perhatikan gambar lingkaran berikut ini.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emyfancybox\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eyellow\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eUNSUR-UNSUR LINGKARAN\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newcommand\\rad\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e Membuat busur lingkaran ---\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"2-create-auxiliary-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-create-auxiliary-lines\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Create auxiliary lines\"\u003e\u003c/a\u003e2. Create auxiliary lines\u003c/h3\u003e\u003cp\u003eNext, in the body of the document to make it easier to draw, first create auxiliary lines.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[step=1cm,gray,very thin]\u003c/span\u003e (-5,-5) grid (5,5);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"3-create-coordinate-axes\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-create-coordinate-axes\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Create Coordinate Axes\"\u003e\u003c/a\u003e3. Create Coordinate Axes\u003c/h3\u003e\u003cp\u003eNext, create the x-axis and y-axis lines using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick,\u0026lt;-\u0026gt;]\u003c/span\u003e (-5,0) -- (5,0)node[anchor=north west] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eX\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick,\u0026lt;-\u0026gt;]\u003c/span\u003e (0,-5) -- (0,5)node[anchor=south east] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eY\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"4-create-a-scale-on-the-axis\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-create-a-scale-on-the-axis\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Create a Scale on the Axis\"\u003e\u003c/a\u003e4. Create a Scale on the Axis\u003c/h3\u003e\u003cp\u003eThen create the x-axis and y-axis values ​​using the following command\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-5,-4,-3,-2,-1,1,2,3,4,5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e cm,1pt) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e cm,-1pt)node[anchor=north] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\x\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-5,-4,-3,-2,-1,1,2,3,4,5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1pt,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e cm) -- (-1pt,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e cm) node[anchor=east] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"5-drawing-circles\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-drawing-circles\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Drawing Circles\"\u003e\u003c/a\u003e5. Drawing Circles\u003c/h3\u003e\u003cp\u003eIt\u0026rsquo;s time to draw a circle with a radius of 4 units via the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [line width=1pt] circle (4);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe results from steps one to five look like the following image.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUsQOtZW6YtfWlD0k7cK_zHdAh0z9WLrRzh4enSmGv6-VP81pHC4CeVLJQJaHIitYuKcZ6jKiSq2Z5oSrWJz2zqiNDinZn7x24SaeM5koeT-kiTF282kGH7AqXZbTBP1Z4PoNdzPrBcoQ/s1100/unsurling1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n\u003cimg alt=\"mengambar lingkaran\" border=\"0\" data-original-height=\"526\" data-original-width=\"1100\" height=\"153\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUsQOtZW6YtfWlD0k7cK_zHdAh0z9WLrRzh4enSmGv6-VP81pHC4CeVLJQJaHIitYuKcZ6jKiSq2Z5oSrWJz2zqiNDinZn7x24SaeM5koeT-kiTF282kGH7AqXZbTBP1Z4PoNdzPrBcoQ/w320-h153/unsurling1.png\" title=\"mengambar lingkaran\" width=\"320\" /\u003e\n\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"6-determine-coordinate-points\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-determine-coordinate-points\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Determine Coordinate Points\"\u003e\u003c/a\u003e6. Determine Coordinate Points\u003c/h3\u003e\u003cp\u003eNext, create the center point of the circle using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [fill=blue] circle (3pt);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIt\u0026rsquo;s a good idea to first create the coordinate points that will be used to draw the radius, diameter, chord, chord and apothem. These points are exampled as follows.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=above:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{A}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (A) at (-3,2.8);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=left:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{B}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (B) at (-4,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=right: \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{C}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (C) at (4,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=above:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{D}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (D) at (0,4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=right:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{E}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (E) at (2.8,3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{F}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (F) at (0,-4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=left:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{G}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (G) at (2,-2);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{H}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (H) at (3,-3);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"7-draw-the-fingers\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-draw-the-fingers\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Draw the Fingers\"\u003e\u003c/a\u003e7. Draw the Fingers\u003c/h3\u003e\u003cp\u003eThen from the center of the circle we will create radius OA via the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [line width=1.5pt,red] (0,0) -- (-2.8,2.8);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"8-drawing-diameter\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-drawing-diameter\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Drawing Diameter\"\u003e\u003c/a\u003e8. Drawing Diameter\u003c/h3\u003e\u003cp\u003eNext, to create a diameter, for example diameter BC, use the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [line width=1.5pt,red] (-4,0) -- (4,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=left:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{B}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (B) at (-4,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=right: \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{C}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (C) at (4,0);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"9-drawing-slopes-and-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#9-drawing-slopes-and-angles\" class=\"header-mark\" aria-label=\"Direct link to section: 9. Drawing Slopes and Angles\"\u003e\u003c/a\u003e9. Drawing Slopes and Angles\u003c/h3\u003e\u003cp\u003eNext we will draw a circular mesh, for example an ODE mesh drawn using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[black,fill=red!85]\u003c/span\u003e (O)--(D) node[above,pos=0.5] \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e arc (90:45:4) -- cycle;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the ODE mesh, we will create an angle, for example called angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. The command to write the angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is as follows.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pic\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=blue,angle radius=.7cm,pic text=$\\alpha$,angle eccentricity=.65]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eangle=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eE--O--D\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"10-drawing-bowstrings\" class=\"headerLink\"\u003e\n \u003ca href=\"#10-drawing-bowstrings\" class=\"header-mark\" aria-label=\"Direct link to section: 10. Drawing Bowstrings\"\u003e\u003c/a\u003e10. Drawing Bowstrings\u003c/h3\u003e\u003cp\u003eNext, draw the CF chord, this chord is needed to create the apothem and segment lines. The CF chord is drawn using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [line width=1.5pt,red] (0,-4) -- (4,0);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"11-drawing-apothem\" class=\"headerLink\"\u003e\n \u003ca href=\"#11-drawing-apothem\" class=\"header-mark\" aria-label=\"Direct link to section: 11. Drawing Apothem\"\u003e\u003c/a\u003e11. Drawing Apothem\u003c/h3\u003e\u003cp\u003eNext draw the OG apothem. So from the center point of the circle O(0,0) we will draw a line perpendicular to the chord CF. The command is as follows.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [line width=1.5pt,blue] (0,0) -- (2,-2);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd, because the apothem OG is perpendicular to the chord CF, to create a right angle sign at G, we make the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\siku\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eC\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eG\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eO\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"12-drawing-sections\" class=\"headerLink\"\u003e\n \u003ca href=\"#12-drawing-sections\" class=\"header-mark\" aria-label=\"Direct link to section: 12. Drawing Sections\"\u003e\u003c/a\u003e12. Drawing Sections\u003c/h3\u003e\u003cp\u003eNext is drawing the sections. A segment is an area bounded by a chord and a circular arc. For example, the section created is the CFH section. The drawing command is as follows.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[black,fill=green!85]\u003c/span\u003e (C)--(F) node[above,pos=0.5] \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e arc (-90:0:4) -- cycle;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"13-drawing-a-circle-arc\" class=\"headerLink\"\u003e\n \u003ca href=\"#13-drawing-a-circle-arc\" class=\"header-mark\" aria-label=\"Direct link to section: 13. Drawing a Circle Arc\"\u003e\u003c/a\u003e13. Drawing a Circle Arc\u003c/h3\u003e\u003cp\u003eNext is to draw circular arcs AB, BF, CF, CE, and BF.\nTo draw a BD arc, use the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-16\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=2pt, green]\u003c/span\u003e (-\u003cspan class=\"k\"\u003e\\rad\u003c/span\u003e,0) arc (180:90:\u003cspan class=\"k\"\u003e\\rad\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTo draw a BF arc, use the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-17\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=2pt, cyan]\u003c/span\u003e (0,-\u003cspan class=\"k\"\u003e\\rad\u003c/span\u003e) arc (270:180:\u003cspan class=\"k\"\u003e\\rad\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTo draw a CF arc, use the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-18\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=2pt, magenta]\u003c/span\u003e (0,-\u003cspan class=\"k\"\u003e\\rad\u003c/span\u003e) arc (-90:0:\u003cspan class=\"k\"\u003e\\rad\u003c/span\u003e);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003ePlease continue to draw the CD arc 😀. Now you can see the results of steps one to step thirteen as in the following image.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYJlBsp4U2ezegTVhy79YJm739mpD-ixrnLc62r5rac4LMxdjRHOW-hYm0IvAg52Jyqo-LwZCY1eYGjCzTrWNrWEzVMjJiF2ZhVV4pGjTnVEkICoCr-HVH1F2mYm7D5p3V_ovCuXLd_Bc/s1190/unsurling2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n\u003cimg alt=\"unsur-unsur lingkaran\" border=\"0\" data-original-height=\"491\" data-original-width=\"1190\" height=\"132\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYJlBsp4U2ezegTVhy79YJm739mpD-ixrnLc62r5rac4LMxdjRHOW-hYm0IvAg52Jyqo-LwZCY1eYGjCzTrWNrWEzVMjJiF2ZhVV4pGjTnVEkICoCr-HVH1F2mYm7D5p3V_ovCuXLd_Bc/w320-h132/unsurling2.png\" title=\"unsur-unsur lingkaran\" width=\"320\" /\u003e\n\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"final-results-and-usage-examples\" class=\"headerLink\"\u003e\n \u003ca href=\"#final-results-and-usage-examples\" class=\"header-mark\" aria-label=\"Direct link to section: Final Results and Usage Examples\"\u003e\u003c/a\u003eFinal Results and Usage Examples\u003c/h3\u003e\u003cp\u003eFinally we will remove the auxiliary lines, the x and y axis lines, as well as the x and y axis coordinate values. You can delete the command directly or insert a % sign in front of the command, so that it is not read by LaTeX. Look at the following image.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpBjqSZodnknsmiLLd8W3TkJ_s0ULb23A9HpCvZUBSz3n1DASG49Rw0v0c-UxeF-5fM7bH_fSV0cJaU92dXC7hIYq15vgJ-jX4UCnEQlmoE6JIcQUaEkr6X1Ns1GCiJ7Svi8ejVwH0nuk/s1218/unsurling3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n\u003cimg alt=\"unsur-unsur lingkaran\" border=\"0\" data-original-height=\"486\" data-original-width=\"1218\" height=\"128\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpBjqSZodnknsmiLLd8W3TkJ_s0ULb23A9HpCvZUBSz3n1DASG49Rw0v0c-UxeF-5fM7bH_fSV0cJaU92dXC7hIYq15vgJ-jX4UCnEQlmoE6JIcQUaEkr6X1Ns1GCiJ7Svi8ejVwH0nuk/w320-h128/unsurling3.png\" title=\"unsur-unsur lingkaran\" width=\"320\" /\u003e\n\u003c/a\u003e\nIf you are an educator and want to use the figure as worksheet material, you can add a few questions about circle elements below the image.\u003c/p\u003e\n\u003cp\u003eThis is a guide to drawing circle elements using LaTeX. If any part needs correction or clarification, please leave a note in the comments.\u003c/p\u003e\n", "language": "en" }, { "title" : "Drawing Flat Shapes", "date_published" : "2025-12-14T13:08:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-bangun-datar/", "url" : "https://www.aantriono.com/en/menggambar-bangun-datar/", "summary": " Drawing Figures with TikZ in LaTeXGambar can be used as a medium to convey messages from the maker to other people. Many applications can be used to draw an object. LaTeX is no exception, there are several packages available that are used to draw in the documents you create, such as the PGF, TikZ, and PSTricks packages. Of course, the drawing process begins by writing the package in the preamble, and certain commands in the preamble or in the body of the document, then compiling it so that the results of the drawing commands are visible. So how do you draw in LaTeX?\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-figures-with-tikz-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-figures-with-tikz-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Figures with TikZ in LaTeX\"\u003e\u003c/a\u003eDrawing Figures with TikZ in LaTeX\u003c/h2\u003e\u003cp\u003eGambar can be used as a medium to convey messages from the maker to other people. Many applications can be used to draw an object. LaTeX is no exception, there are several packages available that are used to draw in the documents you create, such as the PGF, TikZ, and PSTricks packages. Of course, the drawing process begins by writing the package in the preamble, and certain commands in the preamble or in the body of the document, then compiling it so that the results of the drawing commands are visible.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr1RzgUG387-VUE4TAPKEgZYJc2BSoHAe5qTUvHs2iIH2qi9V1oqX_l7YIJSvS9RymBFwwMmfIAinUQdOy1Jltd4avUYjAWHblmTy4V56NdaydVztLyLQ74wH45za0On7vFbXnQUwkZZM/w200-h172/menggambar10.png' alt=\"menggambar lingkaran di latex\" \u003e\nSo how do you draw in LaTeX?\u003c/p\u003e\n\u003ch3 id=\"getting-to-know-the-tikz-package\" class=\"headerLink\"\u003e\n \u003ca href=\"#getting-to-know-the-tikz-package\" class=\"header-mark\" aria-label=\"Direct link to section: Getting to know the TikZ Package\"\u003e\u003c/a\u003eGetting to know the TikZ Package\u003c/h3\u003e\u003cp\u003eIn this article, I will provide a simple guide for drawing several flat shapes using the TikZ package. You must first include the package \\usepackage{TikZ} in the preamble to the document. By using the TikZ package we can place images in inline mode or in command scope (environment). Look at the following image to see the difference.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsI_ORtdVC-Gl9DCpbpzxyzEYg_iiWnC2G6GIRBeVFHcK8-Acju26lxj_3x8_2MCcflAxFuNA08WK9NqoJlS3xrYax7QucNJLS4NC06Nw3EmYr_NUd0RUtL7d2uF42P3JgZYTRU-pKJLc/w320-h102/menggambar1.png' alt=\"menggambar di latex\" \u003e\nThe images of circles, triangles, rectangles and ellipses are obtained from commands such as the following.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar garis-garis bantu (grid) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[step=1cm,gray,very thin]\u003c/span\u003e (-4,-4) grid (4,4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar garis sumbu X ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick,\u0026lt;-\u0026gt;]\u003c/span\u003e (-5,0) -- (5,0)node[anchor=north west] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eX\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar garis sumbu Y ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick,\u0026lt;-\u0026gt;]\u003c/span\u003e (0,-5) -- (0,5)node[anchor=south east] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eY\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar nilai sumbu X ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-4,-3,-2,-1,1,2,3,4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e cm,1pt) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e cm,-1pt)node[anchor=north] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\x\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar nilai sumbu Y ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-5,-4,-3,-2,-1,1,2,3,4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1pt,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e cm) -- (-1pt,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e cm) node[anchor=east] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e(0,0) circle (2); \u003cspan class=\"c\"\u003e% --- lingkaran berada pada koordinat (0,0) dan berjari-jari 2 ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menempatkan titik O pada koordinat (0,0) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eO\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e](O) at (0:0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Membuat noktah di pusat lingkaran (0,) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [fill=gray] (O) circle (2pt);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eNow we will detail how to draw these shapes.\u003c/p\u003e\n\u003ch3 id=\"1-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-circle\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Circle\"\u003e\u003c/a\u003e1. Circle\u003c/h3\u003e\u003cp\u003eOnce the TikZ package is listed in the preamble, continue by creating the \\begin{tikzpicture} \u0026hellip; \\end{tikzpicture} command scope within the document body. It\u0026rsquo;s a good idea to first create auxiliary lines (grid) and Cartesian coordinates to help determine the coordinates of the image to be created. The command to draw a circle is as seen in the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar garis-garis bantu (grid) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[step=1cm,gray,very thin]\u003c/span\u003e (-4,-4) grid (4,4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar garis sumbu X ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick,\u0026lt;-\u0026gt;]\u003c/span\u003e (-5,0) -- (5,0)node[anchor=north west] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eX\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar garis sumbu Y ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick,\u0026lt;-\u0026gt;]\u003c/span\u003e (0,-5) -- (0,5)node[anchor=south east] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eY\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar nilai sumbu X ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-4,-3,-2,-1,1,2,3,4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e cm,1pt) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e cm,-1pt)node[anchor=north] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\x\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar nilai sumbu Y ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\y\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-5,-4,-3,-2,-1,1,2,3,4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1pt,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e cm) -- (-1pt,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e cm) node[anchor=east] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\y\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e(0,0) circle (2); \u003cspan class=\"c\"\u003e% --- lingkaran berada pada koordinat (0,0) dan berjari-jari 2 ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menempatkan titik O pada koordinat (0,0) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eO\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e](O) at (0:0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Membuat noktah di pusat lingkaran (0,) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [fill=gray] (O) circle (2pt);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting image looks like the following.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEig03aZmQ0ZS7cM6BcqmUXySItzvoPa8BmSU34R4d02qtAiftjJAdjQ2JApq_nH8gP4j-xR5o5A1RDu6lKh15fNkCfuanZR2vkb45EGLDwNsZOOgdXCvJ-rkakyZdtU2uSf88tSGU_qGck/w320-h129/menggambar2.png' alt=\"menggambarlingkaran di latex\" \u003e\nTo color the circle and thicken the circumference of the circle, we add the [ultra thick, color] option after the \\draw command, so that the command to draw the circle becomes \\draw\u003ca href=\"0,0\" rel=\"\"\u003eultra thick,red\u003c/a\u003ecircle (2); . Look at the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkitskq0mt4v3fgmtzI37KaSgnPKYiAfDxTO7LqmGxu92PO7an1HAicBDuOdVXhCltQWUcHpRqkDI4zK-81oiQEYGjmCEdNuNwqCAVDTheNIi60vizIgXinZR4Z0FXSBnI8tAOkw-DptE/w320-h128/menggambar3.png' alt=\"Menggambar lingkaran di LateX\" \u003e\nThen, if you want to give double coloring to the circle, give a command such as: \\shadedraw[ultra thick,draw=blue,inner color=blue,outer color=red] (0,0) circle (2);\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFzkucT_j9VFHYCuhhMFP72xUF4L-JUxY_zFU_OBBTYrWXfaUXzNB_sPApwZLNNzRe4OoAJPObPpU8Z1an1wC0z-gkLmre09VkqDycd5uKeHyibPDmIg3FyFrFQYhwPVX05kJbz2Ks0tM/w320-h128/menggambar4.png' alt=\"Menggambar lingkaran di LateX\" \u003e\u003c/p\u003e\n\u003ch3 id=\"2-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Triangle\"\u003e\u003c/a\u003e2. Triangle\u003c/h3\u003e\u003cp\u003eDrawing a triangle means we connect three lines. For example, in the triangle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, this means we connect a line from point A to point B then to point C then back to point A. If the triangle is a right triangle, then we put a right angle at one of the vertices. Then also mark the angles for the other two corner points. For example, consider the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar segitiga ABC berwarna merah---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [red](0,0)--(2,0)--(0,2)--cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Koordinat titik A terletak pada (0,0) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eA\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (A) at (0,0); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Koordinat titik B terletak pada (2,0) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eB\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (B) at (2,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Koordinat titik c terletak pada (0,2) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=above:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eC\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (C) at (0,2);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar tanda siku di titik A ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue](A) +(3mm,0mm) |- +(0mm,3mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar sudut di titik B ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pic\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=blue,angle radius=.6cm,pic text=\\scriptsize $\\beta$, angle eccentricity=.65]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eangle=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eC--B--A\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar sudut di titik C ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pic\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=blue,angle radius=.6cm,pic text=\\scriptsize $\\gamma$, angle eccentricity=.65]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eangle=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eA--C--B\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is as shown in the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwq4xec-zhPL5_yLKzOS-VVPsTXB7BDcU7Yy4tRR_71I7dBlxitxhuzlbBIP2iswslNHLHRs405M44HqC_rv1Dx-1RLgXoqG15U9U1a8QvllOmqIwHc8y6HHay_RCwgKFAcJI6-i4s2VU/w320-h141/menggambar7.png' alt=\"menggambar segitiga dalam latex\" \u003e\nIf you want to color the triangle, then add the [fill=color] command to the triangle drawing command.\u003c/p\u003e\n\u003ch3 id=\"3-rectangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-rectangle\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Rectangle\"\u003e\u003c/a\u003e3. Rectangle\u003c/h3\u003e\u003cp\u003eLike drawing a triangle, drawing a rectangle means we connect four lines where there are two lines that are the same and parallel. Suppose we draw a rectangle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABCD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, meaning we connect the line from point A to point B then to point C then to point D and back again to point A. For example, consider the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Menggambar persegi panjang ABCD berwarna biru---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue] (0,0) -- (4,0) -- (4,3) -- (0,3) -- cycle;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Koordinat titik A terletak pada (0,0) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eA\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (A) at (0,0); \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Koordinat titik B terletak pada (4,0) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eB\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (B) at (4,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Koordinat titik C terletak pada (4,3) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=above:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eC\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (C) at (4,3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Koordinat titik D terletak pada (0,3) ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=above:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eD\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (D) at (0,3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar diagonal AC ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [red] (0,0) -- (4,3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar diagonal BD ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [red] (4,0) -- (0,3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar tanda siku di titik A ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue](A) +(3mm,0mm) |- +(0mm,2.4mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar tanda siku di titik B ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue](B) +(-3mm,0mm) |- +(0mm,2.4mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar tanda siku di titik C ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue](C) +(-3mm,0mm) |- +(0mm,-2.4mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar tanda siku di titik D ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue](D) +(3mm,0mm) |- +(0mm,-2.4mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document looks like the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlUJoR6JfPss9YipWIyBEyd0K2NJWgQnUZfkUKr5EeImMBtpaia9KcUogPEnaOY2qhl4IE4BO0_U2cMVOe1alk-wttbKrPjlNiZYmhef8KMWRtH4AjSIuH8QikeZb4n8jc4t7g-iQV75Y/w320-h130/menggambar6.png' alt=\"menggambar persegi panjang di latex\" \u003e\nIf you want to color the rectangle, then add the [fill=color] command to the rectangle drawing command.\nApart from this method, there is a simple way to draw a rectangle. You simply connect the coordinates of the lower left corner and the coordinates of the upper right corner by a rectangle. So for the rectangle in the image above, in the third line you can write the command:\n\n \n \\(\\fbox{\\draw (0,0) rectangle (4,3);}\\)\u003c/p\u003e\n\u003ch3 id=\"4-ellipse\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-ellipse\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Ellipse\"\u003e\u003c/a\u003e4. Ellipse\u003c/h3\u003e\u003cp\u003eAn ellipse can be drawn using the circle or ellipse commands. To make it clearer, look at the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Perhatikan perbedaan perintah menggambar elips berikut ini ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Elips berwarna biru ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue](0,0) circle [x radius=2cm, y radius=1cm];\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Elips berwarna merah ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [red](5,0) circle (2 and 1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e%--- Elips berwarna hijau ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [green](10,0) ellipse (2 and 1);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is as shown in the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtHf5SYlkz7bU6t8hK0AuxRaNCDoNREpNhTQJyOt8WxQ_oUDAEEWxJ8S9rDJNI6j4ZvOISukg-vpaSw_XUSdqqoL53XSpXeKKYEXmgfUVGUCaHEEVoUPipoUn8hHkHsB5T1B3ktS4M5js/w320-h87/menggambar8.png' alt=\"gambar elips di latex\" \u003e\u003c/p\u003e\n\u003ch3 id=\"5-parallelogram\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-parallelogram\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Parallelogram\"\u003e\u003c/a\u003e5. Parallelogram\u003c/h3\u003e\u003cp\u003eTo draw a parallelogram, we connect four lines like drawing a rectangle, then on two sides there are two lines made at an angle. To make it clearer, pay attention to the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Mengatur kemiringan garis ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e [xslant=1,xscale=2.5,scale=1.5]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar jajar genjang ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue](0,0) rectangle (1,1)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar diagonal AC ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e coordinate(C) node[above]\u003cspan class=\"nb\"\u003e{\u003c/span\u003eC\u003cspan class=\"nb\"\u003e}\u003c/span\u003e -- (0,0) coordinate(A) node[below]\u003cspan class=\"nb\"\u003e{\u003c/span\u003eA\u003cspan class=\"nb\"\u003e}\u003c/span\u003e (1,0)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"c\"\u003e%--- Menggambar diagonal BD ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e coordinate(B) node[below]\u003cspan class=\"nb\"\u003e{\u003c/span\u003eB\u003cspan class=\"nb\"\u003e}\u003c/span\u003e -- (0,1) coordinate(D) node[above]\u003cspan class=\"nb\"\u003e{\u003c/span\u003eD\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document looks like the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjT54QMEW4_o57132g3lxqNOuQuUbVcw8Ry2yyRTyC-Nm1n0oqCC7WFR5UNyK4bv492h9DTost6yz-gBQAX25YQV6bhPH6A8TICDj1c80jdmtMSnwjH-Yy61NigJxb_j7JFiAMvSoSpcX0/w320-h90/menggambar9.png' alt=\"Menggambar jajar genjang di latex\" \u003e\nThis is a guide on how to draw circles, triangles, rectangles, ellipses and parallelograms in LaTeX using the TikZ package. If there are errors, please correct them, if you don\u0026rsquo;t understand, let\u0026rsquo;s discuss, if you have input, please write them here, if you are happy, I will make another guide. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Creating Question Scripts with LaTeX", "date_published" : "2025-12-14T07:12:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/membuat-naskah-soal/", "url" : "https://www.aantriono.com/en/membuat-naskah-soal/", "summary": " Creating Assessment Question Scripts with LaTeXS It has become a routine and rule that in every school in the middle and end of the semester, the school carries out mid-term assessments and end-of-semester/year assessments. Each teacher also creates questions to test for their students. Usually, if questions created by teachers are used for their own schools, the questions have the following structure.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"creating-assessment-question-scripts-with-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-assessment-question-scripts-with-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Creating Assessment Question Scripts with LaTeX\"\u003e\u003c/a\u003eCreating Assessment Question Scripts with LaTeX\u003c/h2\u003e\u003cp\u003eS\nIt has become a routine and rule that in every school in the middle and end of the semester, the school carries out mid-term assessments and end-of-semester/year assessments. Each teacher also creates questions to test for their students. Usually, if questions created by teachers are used for their own schools, the questions have the following structure.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eLocal government logo and school identity.\u003c/li\u003e\n\u003cli\u003eSeparating line header/logo with type of assessment, name of subject being tested and time of implementation, student identity and other related matters.\u003c/li\u003e\n\u003cli\u003eInstructions for working on questions.\u003c/li\u003e\n\u003cli\u003eQuestion\u003c/li\u003e\n\u003cli\u003ePlace to write the answer (if the question is a fill-in-the-blank form).\u003c/li\u003e\n \u003c/ul\u003e\n\u003ch3 id=\"guide-to-creating-questions-with-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#guide-to-creating-questions-with-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Guide to Creating Questions with LaTeX\"\u003e\u003c/a\u003eGuide to Creating Questions with LaTeX\u003c/h3\u003e\u003cp\u003eFurthermore, on this occasion I will provide a simple guide on creating a class VIII and IX mid-semester assessment question paper written using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. How to. Follow the following guide.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eOpen your Tex editor.\u003c/li\u003e\n\u003cli\u003ePrepare your school/local government logo. Put it in the same folder as you wrote the question.\u003c/li\u003e\n\u003cli\u003eNext, in making this question text, I will use legal size paper, font size \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e11\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e pt, and margins \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, fourier writing type and fill-in type questions which are divided into two (\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e) columns and the questions created are mathematics subject questions. Write down all the packages needed in the preamble to the document as follows.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[legal, 11pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[indonesian]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efourier\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% --- jenis huruf ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=1cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% --- batas margin ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earray, multicol\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% --- untuk membuat kolom ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% --- menuliskan persamaan matematika ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[shortlabels]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumitem\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% --- membuat daftar butir ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% --- menyisipkan gambar ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eadjustbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"c\"\u003e% --- menempatkan gambar dan teks dalam minipage ---\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e................\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eThen you insert your local government logo and the text of your school name using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\adjustbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003evalign=c\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.1\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=4cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003elogolamteng\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\hfill\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\adjustbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003evalign=c\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.8\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\t\u003cspan class=\"k\"\u003e\\large\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003ePEMERINTAH KABUPATEN LAMPUNG TENGAH\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\tDINAS PENDIDIKAN DAN KEBUDAYAAN\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\tUPTD SATUAN PENDIDIKAN SMP NEGERI PURWOREJO\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\tKECAMATAN PADANG RATU\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\small\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003eAlamat : Jln. Perintis Rogowungu Purworejo Kec. Padang Ratu Kab. Lampung Tengah 34176\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\t\t\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\hfil\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is as shown below.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2SItLuBbun4JuCRRiqf4s6wadZqctK7ZO-XLdgI_f6EsGXC3KdbKqrdU2Kjm1I9YSf2pc-oSfMr7QtcJWXPYr21uHOHcvJp4ugN7cphftZUzQXv4DlAcDuUAqr2Ehxb9E2hAmjaUA50w/s1233/pts1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n\u003cimg alt=\"cara membuat soal di latex\" border=\"0\" data-original-height=\"477\" data-original-width=\"1233\" height=\"124\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2SItLuBbun4JuCRRiqf4s6wadZqctK7ZO-XLdgI_f6EsGXC3KdbKqrdU2Kjm1I9YSf2pc-oSfMr7QtcJWXPYr21uHOHcvJp4ugN7cphftZUzQXv4DlAcDuUAqr2Ehxb9E2hAmjaUA50w/w320-h124/pts1.png\" title=\"cara membuat soal di latex\" width=\"320\" /\u003e\n\u003c/a\u003e\n5. To create a line separating the school logo and header with the type of assessment, subject identity, student identity, and so on, in this case \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e lines of different thickness will be created, use the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\vspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e-3mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hrule\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\vspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e.75mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hrule\u003c/span\u003e height 1.5pt\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eNext, the title of the assessment carried out by the school will be created, the identity of the subject being tested along with the identity of the student and other related matters. Use the command as follows.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ePENILAIAN TENGAH SEMESTER (PTS)\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSEMESTER GENAP TAHUN PELAJARAN 2020/2021\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\vspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e.2cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[!ht]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\small\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003elclp\u003cspan class=\"nb\"\u003e{\u003c/span\u003e4.5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003elcl\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\tMata Pelajaran\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e:\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003eMatematika\u003cspan class=\"nb\"\u003e\u0026amp;\u0026amp;\u003c/span\u003eHari/Tanggal\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e:\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003eSelasa, 2 Maret 2021 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\tNama\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e:\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e..........\u003cspan class=\"nb\"\u003e\u0026amp;\u0026amp;\u003c/span\u003eKelas\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e:\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003eIX ...\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\tNo.Peserta Ujian\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e:\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e..........\u003cspan class=\"nb\"\u003e\u0026amp;\u0026amp;\u003c/span\u003eWaktu\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e:\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e07.30-09.30\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is as shown below.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEd36efdU4FrZ8vXsG9WPT2Sx78s20Xq_t6l46bjUbgSVXt5hyphenhyphen2Wem49S0iC25kwSLJUQdT4RnwSnNpAS4mBX9XjiY5bVpu-M2jO4oAsA7bkAwLWqIHxvACHZw3pKPb9-WhzTL4dQ77ho/s1226/pts2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n\u003cimg alt=\"membuat soal di latex\" border=\"0\" data-original-height=\"409\" data-original-width=\"1226\" height=\"107\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEd36efdU4FrZ8vXsG9WPT2Sx78s20Xq_t6l46bjUbgSVXt5hyphenhyphen2Wem49S0iC25kwSLJUQdT4RnwSnNpAS4mBX9XjiY5bVpu-M2jO4oAsA7bkAwLWqIHxvACHZw3pKPb9-WhzTL4dQ77ho/w320-h107/pts2.png\" title=\"membuat soal di latex\" width=\"320\" /\u003e\n\u003c/a\u003e\n7. Next, instructions for working on the questions will be provided. Suppose the text is as shown in the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[A.]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e \u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ePetunjuk Soal\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[1.]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Mulailah dan akhiri dengan berdoa agar dimudahkan segala urusan!\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Tulislah terlebih dahulu identitasmu pada tempat yang telah disediakan!\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Tuliskan jawabanmu dengan singkat, jelas, dan benar pada tempat yang telah disediakan!\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eThe next step is to create questions. For example, the problem is in the form of filling in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e questions and dividing them into \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e columns. Pay attention to the following code.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e \u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSoal\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\setlength\\columnseprule\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emulticols\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[1.]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Sebuah tabung mempunyai tinggi \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e21\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm dan jari-jari alasnya \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e6\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm. Tentukan luas selimut tabung tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Sebuah tabung mempunyai tinggi \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e13\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm dan jari-jari alasnya \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e7\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm. Tentukan luas sisi tabung. \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Volume sebuah tabung adalah \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e20\u003c/span\u003e\u003cspan class=\"nb\"\u003e.\u003c/span\u003e\u003cspan class=\"m\"\u003e790\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ \u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ecm^\u003c/span\u003e\u003cspan class=\"m\"\u003e3\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e. Jika tinggi tabung tersebut \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e15\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm, tentukan panjang jari-jari tabung \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Sebuah kerucut mempunyai panjang jari-jari alasnya \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e6\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm dan tingginya \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e8\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm. Hitunglah luas sisi kerucut \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Hitunglah luas sisi kerucut yang jari-jarinya \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e7\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm dan garis pelukisnya \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e14\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Jika luas selimut kerucut adalah \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e113\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"m\"\u003e04\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ \u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ecm^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e dan jari-jarinya \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e4\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm. Tentukan luas permukaan kerucut tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Suatu kerucut mempunyai jari-jari \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e30\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e mm, dan tinggi \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e5\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e mm. Tentukan volume kerucut tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Jika luas permukaan suatu bola \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e154\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\ \u003c/span\u003e\u003cspan class=\"nb\"\u003ecm^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e, tentukan panjang jari-jari bola tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Sebuah bangun berbentuk belahan bola padat memiliki jari-jari \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e10\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm. Tentukan luas permukaan bangun tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Sebuah bola berjari-jari \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e6\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm dimasukkan ke dalam tabung berisi air sehingga permukaan di dalam tabung naik. Jika \n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e jari-jari alas tabung \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e15\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e cm, tentukan kenaikan air dalam tabung tersebut.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emulticols\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is as shown below.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNoEUWDUbNj8Kmqs4UdPzD4js0QpwNYLaaF3kMP5IzNobTc7EVGPzLINOdvR3U_-2tq5snwUbY-mMSNepCRP2TP-SnTtAM2RElTXYf36ehlq8Hd_Z1_i1yPH9kVocffIqXIWCeZzcqW2s/s1248/pts3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n\u003cimg alt=\"membuat soal di latex\" border=\"0\" data-original-height=\"489\" data-original-width=\"1248\" height=\"125\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNoEUWDUbNj8Kmqs4UdPzD4js0QpwNYLaaF3kMP5IzNobTc7EVGPzLINOdvR3U_-2tq5snwUbY-mMSNepCRP2TP-SnTtAM2RElTXYf36ehlq8Hd_Z1_i1yPH9kVocffIqXIWCeZzcqW2s/w320-h125/pts3.png\" title=\"membuat soal di latex\" width=\"320\" /\u003e\n\u003c/a\u003e\n9. The final part is to write the answer text, so that students know where to write the answer. Pay attention to the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\item\u003c/span\u003e \u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eJawaban\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e....................................................................................................................................................................................................................................................................................\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"example-results\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-results\" class=\"header-mark\" aria-label=\"Direct link to section: Example Results\"\u003e\u003c/a\u003eExample Results\u003c/h3\u003e\u003cp\u003eAs an example, Below is a display of questions that have been created using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and used in the Mid-Even Semester Assessment this Academic Year.\u003c/p\u003e\n\u003ciframe height=\"480\" src=\"https://drive.google.com/file/d/1vZ-BnQB6de1w6gTG_hDHzD5Aef2s4Dj3/preview\" width=\"640\"\u003e\u003c/iframe\u003e\nThis is a guide to creating a mid-semester or end-of-semester/year assessment question paper written using $\\LaTeX$. I hope this is helpful.\n", "language": "en" }, { "title" : "Clock Animation", "date_published" : "2025-12-14T06:49:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/03/animasi-jam/", "url" : "https://www.aantriono.com/2021/03/animasi-jam/", "summary": " Creating Analog Clock Animations with GeoGebraTime is the entire series covering when processes, actions, or circumstances exist or take place. Units of time are denoted by hours, minutes, and seconds. With a clock we can measure when an event occurs. Most clocks in circulation today are either analog or digital. Analog clocks have three hands. The longest hand shows seconds, the short hand shows hours, and the remaining hand shows minutes. Look at the following image.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"creating-analog-clock-animations-with-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-analog-clock-animations-with-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Creating Analog Clock Animations with GeoGebra\"\u003e\u003c/a\u003eCreating Analog Clock Animations with GeoGebra\u003c/h2\u003e\u003cp\u003eTime is the entire series covering when processes, actions, or circumstances exist or take place. Units of time are denoted by hours, minutes, and seconds. With a clock we can measure when an event occurs. Most clocks in circulation today are either analog or digital. Analog clocks have three hands. The longest hand shows seconds, the short hand shows hours, and the remaining hand shows minutes. Look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs-aObfCTplFeXZKmTaZX6bQJHfe0hUoD0H_BEFqjU8UEazcwXvqg_5RrPB5iCAnPw1j6roTsBZ3vHrJjRsn3L9-HEOoR84OoHQTtKdOmxgXeXQv94bG5fT1hyQFbHW7XtPLMkhjp-ppU/w320-h288/anjam0.png\" title=\"making a clock in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs-aObfCTplFeXZKmTaZX6bQJHfe0hUoD0H_BEFqjU8UEazcwXvqg_5RrPB5iCAnPw1j6roTsBZ3vHrJjRsn3L9-HEOoR84OoHQTtKdOmxgXeXQv94bG5fT1hyQFbHW7XtPLMkhjp-ppU/w320-h288/anjam0.png\" data-sub-html=\"\u003ch2\u003emaking a clock in GeoGebra\u003c/h2\u003e\u003cp\u003emaking a clock in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs-aObfCTplFeXZKmTaZX6bQJHfe0hUoD0H_BEFqjU8UEazcwXvqg_5RrPB5iCAnPw1j6roTsBZ3vHrJjRsn3L9-HEOoR84OoHQTtKdOmxgXeXQv94bG5fT1hyQFbHW7XtPLMkhjp-ppU/w320-h288/anjam0.png' alt=\"making a clock in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003emaking a clock in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eIn this post, I will share a guide on how to draw and animate an analog clock using GeoGebra. With this animation, the hands of the drawn clock will move just like a real clock. How? Follow the guide below.\u003c/p\u003e\n\u003ch3 id=\"steps-to-create-a-clock-animation\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-a-clock-animation\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Create a Clock Animation\"\u003e\u003c/a\u003eSteps to Create a Clock Animation\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen GeoGebra.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSelect the Circle with Centre Through Point tool. Click Circle : centre \u0026amp; radius. In the Graphics View, create circles with radii 5, 4.8, 4, 3, and 0. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_jGO9dyEWUt-Oqz7mhXQk3ziFGF7HmeXH9MEB-yeNvkvwMfs7qxAUIviSqETMuJwa4IVcsa1jgo6HlTDu_nmSQIj1LIG-f_koFE_qg9KBAVP13UIXmf5YWvEOojl6Mlgr6GcqaOPecck/w320-h221/anjam.png\" title=\"Circles in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_jGO9dyEWUt-Oqz7mhXQk3ziFGF7HmeXH9MEB-yeNvkvwMfs7qxAUIviSqETMuJwa4IVcsa1jgo6HlTDu_nmSQIj1LIG-f_koFE_qg9KBAVP13UIXmf5YWvEOojl6Mlgr6GcqaOPecck/w320-h221/anjam.png\" data-sub-html=\"\u003ch2\u003eCircles in GeoGebra\u003c/h2\u003e\u003cp\u003eCircles in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_jGO9dyEWUt-Oqz7mhXQk3ziFGF7HmeXH9MEB-yeNvkvwMfs7qxAUIviSqETMuJwa4IVcsa1jgo6HlTDu_nmSQIj1LIG-f_koFE_qg9KBAVP13UIXmf5YWvEOojl6Mlgr6GcqaOPecck/w320-h221/anjam.png' alt=\"Circles in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCircles in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eNext, click the Slider tool. Click on angle, fill in α, Min=0, Max=84600, increment=1. Click OK. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXO6su6WM87liS-lB408u0p13dEKErpci-YVusRqeJmIjlveq3oEzlutoUsb2IdCeVQY3V800KTaaE_eGQmDtcpatY3AObJ3tVP7cbRblKFcvoInfyY4Kuh1fK8_HYV4wCF5K2OrvMAzY/w320-h229/anjam1.png\" title=\"Slider in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXO6su6WM87liS-lB408u0p13dEKErpci-YVusRqeJmIjlveq3oEzlutoUsb2IdCeVQY3V800KTaaE_eGQmDtcpatY3AObJ3tVP7cbRblKFcvoInfyY4Kuh1fK8_HYV4wCF5K2OrvMAzY/w320-h229/anjam1.png\" data-sub-html=\"\u003ch2\u003eSlider in GeoGebra\u003c/h2\u003e\u003cp\u003eSlider in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXO6su6WM87liS-lB408u0p13dEKErpci-YVusRqeJmIjlveq3oEzlutoUsb2IdCeVQY3V800KTaaE_eGQmDtcpatY3AObJ3tVP7cbRblKFcvoInfyY4Kuh1fK8_HYV4wCF5K2OrvMAzY/w320-h229/anjam1.png' alt=\"Slider in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSlider in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eClick the Point tool. In the graphics view, click on the position of the second circle, just below the number 5, and at the center of the circle. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOV4KGn55NTLgZyRbm0jkx0nnw3RLYT0QlLgmw6QXf_VyOMoqXP84ScW8aScIB3IBLWWBAizfWLPXaTMRwgD_DtSE1Tw8rmpOTLlurCs_nuKaPCbDHJuhQAe15qQX5vZiWlu5_IDXsQyg/w320-h219/anjam2.png\" title=\"Point in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOV4KGn55NTLgZyRbm0jkx0nnw3RLYT0QlLgmw6QXf_VyOMoqXP84ScW8aScIB3IBLWWBAizfWLPXaTMRwgD_DtSE1Tw8rmpOTLlurCs_nuKaPCbDHJuhQAe15qQX5vZiWlu5_IDXsQyg/w320-h219/anjam2.png\" data-sub-html=\"\u003ch2\u003ePoint in GeoGebra\u003c/h2\u003e\u003cp\u003ePoint in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOV4KGn55NTLgZyRbm0jkx0nnw3RLYT0QlLgmw6QXf_VyOMoqXP84ScW8aScIB3IBLWWBAizfWLPXaTMRwgD_DtSE1Tw8rmpOTLlurCs_nuKaPCbDHJuhQAe15qQX5vZiWlu5_IDXsQyg/w320-h219/anjam2.png' alt=\"Point in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003ePoint in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eClick the Line tool. In the graphics view, click on points A and B. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDXIcSdHzQSUbE11-YRb6UfjsleSGyjdUAAYCYxA_vH8i6hm8n0QEBqM5FaUp1x9ohe8cTo4s3ycFje3jIwPmwv-tM_xTEdsCg6HIyVaMY15BomMgEDzudJXqkQUjse2PlkKbDdNvnkko/w320-h222/anjam3.png\" title=\"Line in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDXIcSdHzQSUbE11-YRb6UfjsleSGyjdUAAYCYxA_vH8i6hm8n0QEBqM5FaUp1x9ohe8cTo4s3ycFje3jIwPmwv-tM_xTEdsCg6HIyVaMY15BomMgEDzudJXqkQUjse2PlkKbDdNvnkko/w320-h222/anjam3.png\" data-sub-html=\"\u003ch2\u003eLine in GeoGebra\u003c/h2\u003e\u003cp\u003eLine in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDXIcSdHzQSUbE11-YRb6UfjsleSGyjdUAAYCYxA_vH8i6hm8n0QEBqM5FaUp1x9ohe8cTo4s3ycFje3jIwPmwv-tM_xTEdsCg6HIyVaMY15BomMgEDzudJXqkQUjse2PlkKbDdNvnkko/w320-h222/anjam3.png' alt=\"Line in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eLine in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIn the Input bar, type: \u003ccode\u003eSequence(Rotate(B, α⁰, A), α, 0, 360, 30)\u003c/code\u003e. Press Enter. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPY4GEpGUR_a_NSJrq4ZQQph4QitibHOuhYW5jpfl8BGgoKOBKJU_20eaMkLItgs1kjhH6FzPIjqQCBLmhN1N3atkcnh6szvHvMHbk8spfHVd6d3E_mbQP2-jBGvCJc4CU-3L4ZkGigvg/w320-h229/anjam4.png\" title=\"Circles in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPY4GEpGUR_a_NSJrq4ZQQph4QitibHOuhYW5jpfl8BGgoKOBKJU_20eaMkLItgs1kjhH6FzPIjqQCBLmhN1N3atkcnh6szvHvMHbk8spfHVd6d3E_mbQP2-jBGvCJc4CU-3L4ZkGigvg/w320-h229/anjam4.png\" data-sub-html=\"\u003ch2\u003eCircles in GeoGebra\u003c/h2\u003e\u003cp\u003eCircles in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPY4GEpGUR_a_NSJrq4ZQQph4QitibHOuhYW5jpfl8BGgoKOBKJU_20eaMkLItgs1kjhH6FzPIjqQCBLmhN1N3atkcnh6szvHvMHbk8spfHVd6d3E_mbQP2-jBGvCJc4CU-3L4ZkGigvg/w320-h229/anjam4.png' alt=\"Circles in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCircles in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eStill in the Input bar, type: \u003ccode\u003eRotate(h, -6α, A)\u003c/code\u003e. Press Enter. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPV_0GhtBlbjWwPkHV3huSzG6GiIYsgPLVn2dbP9mn1ZJFss8D7NJ0HP8WTBzqw6RfjDNcXyHPWy-HY3e-IKJsmc9ap2xmrMEkq17EayQ-pCFOUnrtSDLSwVG6L1LR5OgSi-xPlSNj-IM/w320-h229/anjam5.png\" title=\"Circles in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPV_0GhtBlbjWwPkHV3huSzG6GiIYsgPLVn2dbP9mn1ZJFss8D7NJ0HP8WTBzqw6RfjDNcXyHPWy-HY3e-IKJsmc9ap2xmrMEkq17EayQ-pCFOUnrtSDLSwVG6L1LR5OgSi-xPlSNj-IM/w320-h229/anjam5.png\" data-sub-html=\"\u003ch2\u003eCircles in GeoGebra\u003c/h2\u003e\u003cp\u003eCircles in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPV_0GhtBlbjWwPkHV3huSzG6GiIYsgPLVn2dbP9mn1ZJFss8D7NJ0HP8WTBzqw6RfjDNcXyHPWy-HY3e-IKJsmc9ap2xmrMEkq17EayQ-pCFOUnrtSDLSwVG6L1LR5OgSi-xPlSNj-IM/w320-h229/anjam5.png' alt=\"Circles in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCircles in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eNext we will create the long hand that shows seconds. Select the Line tool, click vector. In the graphics view, click the line (from step 5) near the center of the circle and drag the line past the 3rd circle. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1U31yN90Xmy6rKlDL5MqsYIO4Ff-9tQxIs9B2iYiUlwBf9eujwIuB5mgQ9eHGlNdqIf6H2SPgz1EsNpGr4m9ogPiPyz6-msJdIKlYzsjWyzCBEhQAXEh0F4bpJt7VPrnSgqvJzY3iCcY/w320-h228/anjam6.png\" title=\"Vector in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1U31yN90Xmy6rKlDL5MqsYIO4Ff-9tQxIs9B2iYiUlwBf9eujwIuB5mgQ9eHGlNdqIf6H2SPgz1EsNpGr4m9ogPiPyz6-msJdIKlYzsjWyzCBEhQAXEh0F4bpJt7VPrnSgqvJzY3iCcY/w320-h228/anjam6.png\" data-sub-html=\"\u003ch2\u003eVector in GeoGebra\u003c/h2\u003e\u003cp\u003eVector in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1U31yN90Xmy6rKlDL5MqsYIO4Ff-9tQxIs9B2iYiUlwBf9eujwIuB5mgQ9eHGlNdqIf6H2SPgz1EsNpGr4m9ogPiPyz6-msJdIKlYzsjWyzCBEhQAXEh0F4bpJt7VPrnSgqvJzY3iCcY/w320-h228/anjam6.png' alt=\"Vector in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eVector in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eRight-click the newly formed line. Select and click Object Properties. Choose the desired color. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-0Ud0m27S61zBC8k429DqZfx1lbA7TgcZh048eHQRtkg725oK_IgQ55-73xzHRaTLIz-aOOf74K7ChS2YRDrI0vWkbSR3KFYh1BXGad2RseQUo7Q4Z4pDzpMCX2VjyWs-uw4nc3S_mZo/w320-h286/anjam7.png\" title=\"Color in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-0Ud0m27S61zBC8k429DqZfx1lbA7TgcZh048eHQRtkg725oK_IgQ55-73xzHRaTLIz-aOOf74K7ChS2YRDrI0vWkbSR3KFYh1BXGad2RseQUo7Q4Z4pDzpMCX2VjyWs-uw4nc3S_mZo/w320-h286/anjam7.png\" data-sub-html=\"\u003ch2\u003eColor in GeoGebra\u003c/h2\u003e\u003cp\u003eColor in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-0Ud0m27S61zBC8k429DqZfx1lbA7TgcZh048eHQRtkg725oK_IgQ55-73xzHRaTLIz-aOOf74K7ChS2YRDrI0vWkbSR3KFYh1BXGad2RseQUo7Q4Z4pDzpMCX2VjyWs-uw4nc3S_mZo/w320-h286/anjam7.png' alt=\"Color in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eColor in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIn the Input bar, type: \u003ccode\u003eRotate(h, -α/10, A)\u003c/code\u003e. Press Enter. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhArW4rKn2r9HxgKBsEQ9HdsGNQPqzafZAK8LECaZP7ioUoKCXWslqowP4QhxKHiwgRdzF7U0Xcxcbxg2BtFAX8M0M5FtXwBMvakWRIisi_vEQNkuTKEPCz8gtlnvS1MIFktmrkle6qCbY/w320-h229/anjam8.png\" title=\"Vector in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhArW4rKn2r9HxgKBsEQ9HdsGNQPqzafZAK8LECaZP7ioUoKCXWslqowP4QhxKHiwgRdzF7U0Xcxcbxg2BtFAX8M0M5FtXwBMvakWRIisi_vEQNkuTKEPCz8gtlnvS1MIFktmrkle6qCbY/w320-h229/anjam8.png\" data-sub-html=\"\u003ch2\u003eVector in GeoGebra\u003c/h2\u003e\u003cp\u003eVector in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhArW4rKn2r9HxgKBsEQ9HdsGNQPqzafZAK8LECaZP7ioUoKCXWslqowP4QhxKHiwgRdzF7U0Xcxcbxg2BtFAX8M0M5FtXwBMvakWRIisi_vEQNkuTKEPCz8gtlnvS1MIFktmrkle6qCbY/w320-h229/anjam8.png' alt=\"Vector in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eVector in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eNext we will create the long hand that shows minutes. Select the Line tool, click vector. In the graphics view, click the line (from step 10) near the center of the circle and drag it past the 2nd circle. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdyOUOz7btUPhrq44Edo7i2E0OYWrcySsE9WOInlH1qSgMHxzsigqSx6Ue4B3OrLBxgsL317U5Sub49FkzBi6moeDyBnUB6DvVEOU1gm1bNtfuEYqBFoJ5nRcXKMm64YEGjoVZYRZlO6s/w320-h232/anjam9.png\" title=\"Vector in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdyOUOz7btUPhrq44Edo7i2E0OYWrcySsE9WOInlH1qSgMHxzsigqSx6Ue4B3OrLBxgsL317U5Sub49FkzBi6moeDyBnUB6DvVEOU1gm1bNtfuEYqBFoJ5nRcXKMm64YEGjoVZYRZlO6s/w320-h232/anjam9.png\" data-sub-html=\"\u003ch2\u003eVector in GeoGebra\u003c/h2\u003e\u003cp\u003eVector in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdyOUOz7btUPhrq44Edo7i2E0OYWrcySsE9WOInlH1qSgMHxzsigqSx6Ue4B3OrLBxgsL317U5Sub49FkzBi6moeDyBnUB6DvVEOU1gm1bNtfuEYqBFoJ5nRcXKMm64YEGjoVZYRZlO6s/w320-h232/anjam9.png' alt=\"Vector in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eVector in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIn the Input bar, type: \u003ccode\u003eRotate(h, -α/120, A)\u003c/code\u003e. Press Enter. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjs7iz46EEHZzwFulpMOY3LNk7wyjqjgRI-l8lcz6J38H0I7_d1fCV0MPQqrnZ1DhQbPJqXsneNkPn_DhHuSjf4xuBC6UUEc2hwJGBemQLhjGIXaDO58k98RDtuyBlmRR8mmlbwBdGzuDM/w320-h229/anjam10.png\" title=\"Line in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjs7iz46EEHZzwFulpMOY3LNk7wyjqjgRI-l8lcz6J38H0I7_d1fCV0MPQqrnZ1DhQbPJqXsneNkPn_DhHuSjf4xuBC6UUEc2hwJGBemQLhjGIXaDO58k98RDtuyBlmRR8mmlbwBdGzuDM/w320-h229/anjam10.png\" data-sub-html=\"\u003ch2\u003eLine in GeoGebra\u003c/h2\u003e\u003cp\u003eLine in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjs7iz46EEHZzwFulpMOY3LNk7wyjqjgRI-l8lcz6J38H0I7_d1fCV0MPQqrnZ1DhQbPJqXsneNkPn_DhHuSjf4xuBC6UUEc2hwJGBemQLhjGIXaDO58k98RDtuyBlmRR8mmlbwBdGzuDM/w320-h229/anjam10.png' alt=\"Line in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eLine in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eNext we will make the short hand that shows hours. Select the Line tool, click vector. In the graphics view, click the line (from step 12) near the center of the circle and drag it toward the 2nd circle. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIYqjZ3H3OUdhVpmW0XhF32jiYWFogOSiP1hFgO4iv7a76EEQxboAkP68vzSXdEC_BJNLixayU_zHKg7dOi6VWXtrqLZLXjswogXFxedfSbFr1KBoG7vNAvINgrnLZnYpk1aIkwTsznps/w320-h221/anjam11.png\" title=\"Vector in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIYqjZ3H3OUdhVpmW0XhF32jiYWFogOSiP1hFgO4iv7a76EEQxboAkP68vzSXdEC_BJNLixayU_zHKg7dOi6VWXtrqLZLXjswogXFxedfSbFr1KBoG7vNAvINgrnLZnYpk1aIkwTsznps/w320-h221/anjam11.png\" data-sub-html=\"\u003ch2\u003eVector in GeoGebra\u003c/h2\u003e\u003cp\u003eVector in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIYqjZ3H3OUdhVpmW0XhF32jiYWFogOSiP1hFgO4iv7a76EEQxboAkP68vzSXdEC_BJNLixayU_zHKg7dOi6VWXtrqLZLXjswogXFxedfSbFr1KBoG7vNAvINgrnLZnYpk1aIkwTsznps/w320-h221/anjam11.png' alt=\"Vector in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eVector in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIn the Input bar, type: \u003ccode\u003eSequence(Rotate(B, α⁰, A), α, 0, 360, 6)\u003c/code\u003e. Press Enter. Look at the following image.\n\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghPeO3xW5gUu4pmEmKFn7oz0JClkZCBtocLWmXsvuE4KW4AqhwjVbtOoqvNdVy-KZcaqc_prkQEQfPIy1W8rampjDB5yl76XvjpbNZaixCTxGUwkgOgg0OWsOQA7ta0-w50annSSqCkZU/w320-h228/anjam12.png\" title=\"Clock in GeoGebra\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghPeO3xW5gUu4pmEmKFn7oz0JClkZCBtocLWmXsvuE4KW4AqhwjVbtOoqvNdVy-KZcaqc_prkQEQfPIy1W8rampjDB5yl76XvjpbNZaixCTxGUwkgOgg0OWsOQA7ta0-w50annSSqCkZU/w320-h228/anjam12.png\" data-sub-html=\"\u003ch2\u003eClock in GeoGebra\u003c/h2\u003e\u003cp\u003eClock in GeoGebra\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghPeO3xW5gUu4pmEmKFn7oz0JClkZCBtocLWmXsvuE4KW4AqhwjVbtOoqvNdVy-KZcaqc_prkQEQfPIy1W8rampjDB5yl76XvjpbNZaixCTxGUwkgOgg0OWsOQA7ta0-w50annSSqCkZU/w320-h228/anjam12.png' alt=\"Clock in GeoGebra\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eClock in GeoGebra\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSelect the slider tool, click Button. In Caption, type START and in the GeoGebra Script box type \u003ccode\u003eStartAnimation(α)\u003c/code\u003e. Create a STOP button with \u003ccode\u003eStartAnimation(false)\u003c/code\u003e in the GeoGebra Script box. Finally, create a RESET button with \u003ccode\u003eStartAnimation(α=0)\u003c/code\u003e.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eNext, tidy up the clock you have finished drawing. To make the look more appealing, apply colors to the second hand, minute hand, hour hand, the dots, and other elements. Hide the 2nd and 3rd circles. Create the numbers 1–12 and place them in their proper positions. You may also add images to make the clock more attractive.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eBelow is an example result of the guide to creating a clock animation in GeoGebra. Click START to begin, click STOP to pause, and click RESET to return the clock to its initial position.\u003c/p\u003e\n\u003ch3 id=\"example-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-result\" class=\"header-mark\" aria-label=\"Direct link to section: Example Result\"\u003e\u003c/a\u003eExample Result\u003c/h3\u003e\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/fqutsj7v/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Clock\" width=\"650px\"\u003e \u003c/iframe\u003e\n\u003cp\u003eThis guide shows how to draw and animate an analog clock in GeoGebra. If you notice anything that should be corrected, or if you have suggestions or questions, please share them in the comments. I hope this tutorial is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "LaTeX Document Background", "date_published" : "2025-12-14T06:43:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/background-dokumen-latex/", "url" : "https://www.aantriono.com/en/background-dokumen-latex/", "summary": "In some documents, the author may want to add a background to create a certain visual mood. In LaTeX, this can be done either with a full-page image or with a simple page color.\nHow to Add a Document Background in LaTeXIn LaTeX, we can add various backgrounds that cover the full page of the document.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eIn some documents, the author may want to add a background to create a certain visual mood. In LaTeX, this can be done either with a full-page image or with a simple page color.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhq-N1Fo1XS5eF0gOrS3KCFJeeFox2A_fXEDPFCbIBueyhLeFiWy-Jx5Cf5Yrz_ZFRIa_M3as5Blhjf5TBSYopgYhbukx7ITAT8insSxX6Md5xvJCM_YhwF0Z9eqGM85aEBTM_sWwrJzWw/s2048/0001+%25289%2529.jpg' alt=\"LaTeX document background\" \u003e\u003c/p\u003e\n\u003ch2 id=\"how-to-add-a-document-background-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-add-a-document-background-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: How to Add a Document Background in LaTeX\"\u003e\u003c/a\u003eHow to Add a Document Background in LaTeX\u003c/h2\u003e\u003cp\u003eIn LaTeX, we can add various backgrounds that cover the full page of the document.\u003c/p\u003e\n\u003ch3 id=\"basic-requirements\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-requirements\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Requirements\"\u003e\u003c/a\u003eBasic Requirements\u003c/h3\u003e\u003cp\u003eBefore starting, note the following:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eInclude the \u003ccode\u003ebackground\u003c/code\u003e package in the document preamble.\u003c/li\u003e\n\u003cli\u003eSave the background image in the same folder as the \u003ccode\u003e.tex\u003c/code\u003e file.\u003c/li\u003e\n\u003cli\u003eWrite the document content as usual.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eBesides images, we can also set a page color using the command \u003ccode\u003e\\pagecolor{color}\u003c/code\u003e.\u003c/p\u003e\n\u003ch2 id=\"steps-to-create-a-background-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-a-background-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Create a Background in LaTeX\"\u003e\u003c/a\u003eSteps to Create a Background in LaTeX\u003c/h2\u003e\u003ch3 id=\"1-open-a-text-editor\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-a-text-editor\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open a Text Editor\"\u003e\u003c/a\u003e1. Open a Text Editor\u003c/h3\u003e\u003cp\u003eOpen your LaTeX editor, either offline or online such as Overleaf.\u003c/p\u003e\n\u003ch3 id=\"2-define-the-document-class\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-define-the-document-class\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Define the Document Class\"\u003e\u003c/a\u003e2. Define the Document Class\u003c/h3\u003e\u003cp\u003eUse the desired document class. In this example, the \u003ccode\u003earticle\u003c/code\u003e class is used.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[a4paper,11pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"3-add-the-required-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-add-the-required-packages\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Add the Required Packages\"\u003e\u003c/a\u003e3. Add the Required Packages\u003c/h3\u003e\u003cp\u003eIn the preamble, include \u003ccode\u003egraphicx\u003c/code\u003e, \u003ccode\u003excolor\u003c/code\u003e, and \u003ccode\u003ebackground\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebackground\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"4-configure-the-background\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-configure-the-background\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Configure the Background\"\u003e\u003c/a\u003e4. Configure the Background\u003c/h3\u003e\u003cp\u003eThen configure the background with the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\backgroundsetup\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e scale=1,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e angle=0,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e opacity=90,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e contents=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=\\paperwidth,height=\\paperheight]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ekertaskuno\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIn this part we can adjust the scale, rotation, and transparency of the background image.\u003c/p\u003e\n\u003ch3 id=\"5-start-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-start-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Start the Document\"\u003e\u003c/a\u003e5. Start the Document\u003c/h3\u003e\u003cp\u003eBegin the document body with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"6-write-the-document-content\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-write-the-document-content\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Write the Document Content\"\u003e\u003c/a\u003e6. Write the Document Content\u003c/h3\u003e\u003cp\u003eWrite the document content as needed. In this example, a song lyric excerpt is used as the sample text.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\Huge\u003c/span\u003e CINTA DAN KALIMAH\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\medskip\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\Large\u003c/span\u003e Iklim\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003everse\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eKasih sayang tak terpadam\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eRindu pun tak terbilang\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eBak malam dan siang\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eHanya engkau kurindu\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Bukan kerna kata-kata\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Cinta yang subur di minda\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Sucinya cintamu\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Di cermin hati\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSalji yang putih hitam mendidih\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eAndai kulupa padamu\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eLangit yang pijar kembali ceria\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eIndah kukecup keningmu\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Dengan ratapan kutelan bisa\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Pahit-manis madah pujangga\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Menguntum rindu seiras wajahmu\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hskip\u003c/span\u003e 8mm Relung cinta dan kalimah\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eTerbuka pintu saujana kemaafan\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eKau tersenyum mengirai mimpi\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSalju terasa bersamamu, kekasih\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMenyulamkan keindahan\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eCinta dan kalimah\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003everse\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"7-end-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-end-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: 7. End the Document\"\u003e\u003c/a\u003e7. End the Document\u003c/h3\u003e\u003cp\u003eClose the document with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"8-compile-and-review-the-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-compile-and-review-the-result\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Compile and Review the Result\"\u003e\u003c/a\u003e8. Compile and Review the Result\u003c/h3\u003e\u003cp\u003eCompile the document and check the result.\u003c/p\u003e\n\u003ch2 id=\"example-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-result\" class=\"header-mark\" aria-label=\"Direct link to section: Example Result\"\u003e\u003c/a\u003eExample Result\u003c/h2\u003e\u003cp\u003eBelow is an example output from the steps above.\u003c/p\u003e\n\u003ciframe height=\"480\" src=\"https://drive.google.com/file/d/1q_e4uPsgShiUxIB3eMQliO1jgV3YH8Pi/preview\" width=\"640\" style=\"max-width: 100%;\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"additional-tip-color-background-and-keep-aspect-ratio\" class=\"headerLink\"\u003e\n \u003ca href=\"#additional-tip-color-background-and-keep-aspect-ratio\" class=\"header-mark\" aria-label=\"Direct link to section: Additional Tip: Color Background and Keep Aspect Ratio\"\u003e\u003c/a\u003eAdditional Tip: Color Background and Keep Aspect Ratio\u003c/h2\u003e\u003cp\u003eIf you want to use a background that does not fully cover the page while preserving the image proportion, use \u003ccode\u003ekeepaspectratio\u003c/code\u003e. You can also add a page background color using \u003ccode\u003e\\pagecolor\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebackground\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\backgroundsetup\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e scale=1,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e angle=0,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e opacity=0.9,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e contents=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=\\paperwidth,height=\\paperheight,keepaspectratio]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eroot.png\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003elipsum\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pagecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue!40\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\lipsum\u003c/span\u003e\u003cspan class=\"na\"\u003e[1-5]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result will look like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgguRonPpKmDG63M7PlCyuwXPIGANVjJxyejwHJPpRIwAjhjw8v-FFf4Zw2XpXg9A6iO8q1LlLDgv5a2_xB5xbBrF0i9CX9eio_ylnN7kzPiF0bAjIRvgzg6WSiuO2p32ll0J5eQ7wOtOQ/s1067/background.png' alt=\"Background in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat is the guide to adding a background to a LaTeX document using the \u003ccode\u003ebackground\u003c/code\u003e package. If you have corrections or suggestions, please leave them in the comments.\u003c/p\u003e\n", "language": "en" }, { "title" : "Perimeter and Area of ​​a Circle", "date_published" : "2025-12-14T06:40:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/keliling-dan-luas-lingkaran/", "url" : "https://www.aantriono.com/en/keliling-dan-luas-lingkaran/", "summary": "As is known, a circle is a simple closed curve which is the locus of points that have the same distance from a certain point. The circumference of a circle is the length of the path taken along the circle from the origin back to the origin. The area of ​​a circle is the area bounded or surrounded by a circular curve. The formula for finding the circumference and area of ​​a circle is:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eAs is known, a circle is a simple closed curve which is the locus of points that have the same distance from a certain point. The circumference of a circle is the length of the path taken along the circle from the origin back to the origin. The area of ​​a circle is the area bounded or surrounded by a circular curve. The formula for finding the circumference and area of ​​a circle is:\u003c/p\u003e\n\u003cp\u003e\\(\\begin{aligned} \\text{Around}\u0026=2\\pi r\\ \\text{and}\\ \\text{Area}=\\pi r^2 \\end{aligned}\\)\u003c/p\u003e\n\u003cp\u003eWe can create animations in GeoGebra to determine the circumference and area of ​​a circle created. How to? Follow the following guide.\u003c/p\u003e\n\u003ch2 id=\"steps-to-create-an-animation-of-the-circumference-and-area-of-a-circle-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-an-animation-of-the-circumference-and-area-of-a-circle-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to create an animation of the circumference and area of ​​a circle in GeoGebra\"\u003e\u003c/a\u003eSteps to create an animation of the circumference and area of ​​a circle in GeoGebra\u003c/h2\u003e\u003ch3 id=\"1-open-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open GeoGebra\"\u003e\u003c/a\u003e1. Open GeoGebra\u003c/h3\u003e\u003cp\u003eOpen the GeoGebra application on your computer or device.\u003c/p\u003e\n\u003ch3 id=\"2-create-a-radius-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-create-a-radius-slider\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Create a Radius Slider\"\u003e\u003c/a\u003e2. Create a Radius Slider\u003c/h3\u003e\u003cp\u003eClick tools *Sliders*. Click on the Graph View and type Radius, min=$1$, max=$4$ (as needed), increment=$0.1$. Click *OK*. Look at the following image.\u003c/p\u003e\n![tool slider di GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRFD7bfPl54Pj02mOXzJxgbP4jkBOoYqGZjVthLJuIxJFS9PCcg7nBZ6fyWcb0PN1vIhQvVQe-yxq44qaN69_o9AEWS81-4rcdt_mJuDwZdsFqwzmOzCYJVqfOrBEoTwPfwx-V1SnqaMY/w320-h165/animasilingkaran1.png)\n### 3. Create a Circle with the Slider Radius\n\u003cp\u003eClick tools *Circle with center \u0026 radius*. Click on Graphic View, a box appears and type *Fingers*. Look at the following image.\u003c/p\u003e\n![radius lingkaran di GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1DMVZDTPbE1Xq25ClSe_Ll8PgYRGRh7RY_KA_TdP3l-gNSkIImyTkGCyEmzYjJCLc6VURzWmde-6DWKtoazgA5Ldkb8cl9-dBkdpzetut2-0Um70BvwydjLhD5uiG4bHhbUWiPts6D6k/w320-h176/animasilingkaran2.png)\n### 4. Create a circumference formula text\n\u003cp\u003eClick tools *Move*. Click *Sliders* choose *Text* and type Perimeter = $2\\pi r$. Click *OK*. Look at the following image.\u003c/p\u003e\n![rumus keliling lingkaran](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK_Usb7It5gJcNDFTeLrbJxnqYgO1Hi4DnpXeUTphZSDCl3Q3VP4-v2s61yfqhc2DpOrbkLXQF8cM9yM7tleVvOonSl3v4IVX87F91M2NIummOUhH2rT4ui7cUV5-WdDmgb1EJ2vUBzps/w320-h264/animasilingkaran3.png)\n### 5. Create a circumference calculation text\n\u003cp\u003eClick again on Graphic View. Type = $2\\times 3.14\\times \\text{Finger}$. Click *OK*. Look at the following image.\u003c/p\u003e\n![perhitungan keliling lingkaran](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSp0GtdC_cD42ZMYKn1LyfPtPlBhx4ziQTdeukMF-wqwuN9qArNphTPmK6oDr5DSA5ZAzdzVxnsht1dWFPBHdN5MPi2g9dmxh3UR_rF3dzHnl-HM0ysoBVqjvCoHtxz6SjxiQu-3IFJDk/w320-h258/animasilingkaran4.png)\n### 6. Create a Perimeter Variable\n\u003cp\u003eIn column *Inputs*, type $K=2*3,14*Radiku$. Click *enter*. Look at the following image.\u003c/p\u003e\n![variabel keliling di GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiK_mrY05M-L0bQhLBw_NWRDzsyCQIPHYjj_9_qtgVvx0LpXcZGDoZSG4b-nhtqCOoxYY2yOCetSYvsV4szvRYPeY9BO8e0D9uHfLPnrJClvvR8vT8wavM2oJG_ZqjWlfxvzR73KI-2Rbg/w305-h320/animasilingkaran5.png)\n### 7. Create Roving Value Text\n\u003cp\u003eClick tools *Sliders*. Choose *Text* and type $= K$. Click *OK*. Look at the following image.\u003c/p\u003e\n![nilai keliling lingkaran](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzYm6WU5oF0bngMn3yeu1c4RD8TZvEJkbdHSgxx8sMEBdMhNtO1OeBB7Lqangb9lSdV-yN_uy1qQ2NJgbIR6cV9tywrb8RekAshLT5mQALQ1xpdorUtdvTN79Zv_Pp5pOESKGk9chP1ng/w320-h262/animasilingkaran6.png)\n### 8. Create Broad Formula Text\n\u003cp\u003eNext we will create text area of ​​the circle. Click tools *Sliders*, choose *Text*. Click on Graph View, and type $\\text{Area}=\\pi r^2$. Click *OK*. Look at the following image.\u003c/p\u003e\n![rumus luas lingkaran](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHBxjYBrmBZjm5yTWORzkK_7UvriNQAZ4FgkXl5k3ZgfHfp30HW6kIGg8DhsLEJ6ejbAsxNNIMFRO6RvZ5-4_Ei-dvoYRxDbLO8jtz2BVTU0VT_LMBGn6m-NN2GNo11jBflPNIWgHjvAU/w320-h263/animasilingkaran7.png)\n### 9. Create Extensive Calculation Text\n\u003cp\u003eStill in Graphics View, type $=3,14\\times \\text{Radier}^2$. Click *OK*. Look at the following image.\u003c/p\u003e\n![perhitungan luas lingkaran](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWbSJ0g3c-i7pzrw6nMHwttg2oxDka-Svi9K98dr_EvktE_MsEyDKvtByGk1rLlRVTJIWhAiEhQn4r_ks2QfJB15P5neadIYNAdhliAnT1AYZ-ashpVlD41tdmRPZ7LgcdTk53RSS4G8k/w320-h262/animasilingkaran8.png)\n### 10. Create Wide Variables\n\u003cp\u003eIn column *Inputs* type $L=3,14*\\text{Finger}\\textasciicircum 2$, click *enter*. Look at the following image.\u003c/p\u003e\n![variabel luas di GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLSy9sBT60gxnFCd0uyGMASJTScvovUYTeJpQdPZHJGsLv7bg2qAEYGb0lmwNQIewXqdKnQFN1vGj3Z9cYilQZ32AxWf8dCgw42SOCQ-qHbnl2GEtFaYk-M9VLKxazEO06PzLI6xQlBiM/w303-h320/animasilingkaran9.png)\n### 11. Create Broad Value Text\n\u003cp\u003eClick tools *Sliders*, choose *Text*. Click on Graph View and type $=L$. Click *OK*.\u003c/p\u003e\n\u003ch3 id=\"12-create-a-check-box-for-traveling-around\" class=\"headerLink\"\u003e\n \u003ca href=\"#12-create-a-check-box-for-traveling-around\" class=\"header-mark\" aria-label=\"Direct link to section: 12. Create a check box for traveling around\"\u003e\u003c/a\u003e12. Create a check box for traveling around\u003c/h3\u003e\u003cp\u003eNext we will create *Check Box*. Click tools *Sliders* choose *check box*. In the caption, type Surrounding. Next, click the small arrow icon below it and select text1, text2, and text\n\u003col start=\"3\"\u003e\n\u003cli\u003eClick \u003cem\u003eApply.\u003c/em\u003e Look at the following image.\u003c/p\u003e\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgecFKU2317uDGBWOI0Lnb4cakTF8rhTJ6oja8VOG-4GmCaAtnyOMLM3JWO8jsCvs-Ks-LJqF3PXLiyk9HrjRqt5qeYa1-fUdrXDly6_qnDifmaifABqZxoo8ADo-VQgNFIyaL_qTV5POc/w320-h162/animasilingkaran10.png' alt=\"cek box untuk keliling di GeoGebra\" \u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"13-create-a-check-box-for-area\" class=\"headerLink\"\u003e\n \u003ca href=\"#13-create-a-check-box-for-area\" class=\"header-mark\" aria-label=\"Direct link to section: 13. Create a Check Box for Area\"\u003e\u003c/a\u003e13. Create a Check Box for Area\u003c/h3\u003e\u003cp\u003eAs in step $12$, in the caption type Area. Next select text4, text5, and text\n\u003col start=\"6\"\u003e\n\u003cli\u003eClick \u003cem\u003eApply\u003c/em\u003e. Look at the following image.\u003c/p\u003e\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX2WktIrP03GiNJLq7PVOUr-pZLWQKYtJh_ZMRHudT9GD-jK0zKNYAt-CjvlYoYwgdwQKnbpSUdO82nY1hAY8ZKcV5V8CicidEnZ9s9_swFOfN1OCdQn7DRYX-c_T4VbLx3U-2JDbN6EU/w320-h163/animasilingkaran11.png' alt=\"cek box untuk luas di GeoGebra\" \u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"14-check-box-test\" class=\"headerLink\"\u003e\n \u003ca href=\"#14-check-box-test\" class=\"header-mark\" aria-label=\"Direct link to section: 14. Check Box Test\"\u003e\u003c/a\u003e14. Check Box Test\u003c/h3\u003e\u003cp\u003eTo test steps $12$ and $13$, please click on the Perimeter and Area boxes to show/hide the perimeter and area texts. Look at the following image.\u003c/p\u003e\n![uji check box di GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9PkXdMCwCiFDz-QUIYfgldzw6J0wtYF71C2bcHNd3k-OTnLKL_37MsbXTWd0k-UqTsMFt3WVmEt_j2J6AxPYZ6McbYt97WuyPlIWazzViJ48ksm3htxikEFi-WDqT5ueJADATvCtMIz0/w320-h248/animasilingkaran12.png)\n### 15. Create an Animated Control Button\n\u003cp\u003eNext we will create buttons to turn on, turn off, and reset the circumference and area of ​​the circle animation. Click tools *Sliders*, choose *Buttons*. In the caption, type *ON* and in the GeoGebra Script section type *StartAnimation[Finger].* Create another button, in the caption type *OFF* and in the GeoGebra Script section type *StartAnimation[false]*. Create another third button, in the caption type RESET and in the GeoGebra Script section type StartAnimation[false] *Radius=1. *Click* OK.*\u003c/p\u003e\n![tombol kontrol animasi di GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCCnJXQAoF8e-BXCI2_A3iEIfaNV1wp2kaOYCVd1FItn86_H9aFIEhnUp7e59ry6Qjt8ldJso5DhreH2SuUjGt3oqLoViosJFOjUggOd4zAxHz8ky_cTel2FplwmbBvAGBzx3X8Csfe6c/w320-h216/animasilingkaran14.png)\n### 16. Tidy Up the Appearance\n\u003cp\u003eThe final part, tidy up the appearance of the slider, circle, check box, On, Off and Reset buttons, color them according to your wishes.\u003c/p\u003e\n\u003ch3 id=\"the-final-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-final-result\" class=\"header-mark\" aria-label=\"Direct link to section: The final result\"\u003e\u003c/a\u003eThe final result\u003c/h3\u003e\u003cp\u003eBelow is an example of an animation of the circumference and area of ​​a circle using GeoGebra. Check/Uncheck the perimeter or area box to display or hide the circumference and area of ​​the circle text. Click the ON button to turn on the animation, OFF to stop the animation, and RESET to return the circle animation to its initial appearance with radius $1$.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/mjdvyccs/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; width: 100%; max-width: 650px;\" title=\"Keliling dan Luas Lingkaran\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThat's the article on how to create an animation of the circumference and area of ​​a circle using GeoGebra. If there are errors, please correct them, if you don't understand, let's discuss, if you have comments, write them here, if you like it, I'll make another guide. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Proving the Area of ​​a Triangle", "date_published" : "2025-12-14T06:34:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pembuktian-luas-segitiga/", "url" : "https://www.aantriono.com/en/pembuktian-luas-segitiga/", "summary": "As is known, a triangle is a flat figure that has three sides and three angles, and the sum of the measures of the three angles is $180^0$. Also understand that to find the area of ​​a triangle, use the formula:\n\\(L\\ \\Delta =\\frac{1}{2}\\ a\\times t\\)\nWhere did this formula come from? Is there a connection between triangles and quadrilaterals? In this article, I will provide a guide on how to prove the area of ​​a triangle using the area of ​​a quadrilateral using the GeoGebra application.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eAs is known, a triangle is a flat figure that has three sides and three angles, and the sum of the measures of the three angles is $180^0$. Also understand that to find the area of ​​a triangle, use the formula:\u003c/p\u003e\n\u003cp\u003e\\(L\\ \\Delta =\\frac{1}{2}\\ a\\times t\\)\u003c/p\u003e\n\u003cp\u003eWhere did this formula come from? Is there a connection between triangles and quadrilaterals? In this article, I will provide a guide on how to prove the area of ​​a triangle using the area of ​​a quadrilateral using the GeoGebra application.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMa94It9m5rer6KN6rdAenYaiqiknSMYoppj-1jMDkyz4w8TdLXka4bUiHPrmAE9PH3RmSFP48eYUwr7-Dqfa5Z5npMT5l3kPGQ-PBL7sTAEJracoww6R0JP5KccMOaA_566kUL-WB4kA/s579/segitiga02.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"rotasi segitiga\" border=\"0\" data-original-height=\"428\" data-original-width=\"579\" height=\"237\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMa94It9m5rer6KN6rdAenYaiqiknSMYoppj-1jMDkyz4w8TdLXka4bUiHPrmAE9PH3RmSFP48eYUwr7-Dqfa5Z5npMT5l3kPGQ-PBL7sTAEJracoww6R0JP5KccMOaA_566kUL-WB4kA/w320-h237/segitiga02.png\" title=\"rotasi segitiga\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch2 id=\"steps-to-prove-the-area-of-a-triangle-using-the-area-of-a-quadrilateral-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-prove-the-area-of-a-triangle-using-the-area-of-a-quadrilateral-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Prove the Area of ​​a Triangle Using the Area of ​​a Quadrilateral in GeoGebra\"\u003e\u003c/a\u003eSteps to Prove the Area of ​​a Triangle Using the Area of ​​a Quadrilateral in GeoGebra\u003c/h2\u003e\u003cp\u003eOK, just take a look at the following guide.\u003c/p\u003e\n\u003ch3 id=\"1-open-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open GeoGebra\"\u003e\u003c/a\u003e1. Open GeoGebra\u003c/h3\u003e\u003cp\u003eOpen the GeoGebra application on your computer or device.\u003c/p\u003e\n\u003ch3 id=\"2-create-a-sliderlaunch\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-create-a-sliderlaunch\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Create a Slider/Launch\"\u003e\u003c/a\u003e2. Create a Slider/Launch\u003c/h3\u003e\u003cp\u003eClick tools *Glide*, click anywhere on the Graphic Display and a box will appear, click Corner, fill in as below. Click OK. Look at the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVG1ASsWq6o07U19DkHPeH14fILRFX2qe3jBear7Uja94xLS-_t0WB4tWT6HrsuA4zgyR5YHpFRJqHzO-F-4ff1-cwtlgleNoQYFimyRUQymhe3blkN9MfOKYo-lboKcgP38ZYwGpoyF0/s883/segitiga03.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"slider di GeoGebra\" border=\"0\" data-original-height=\"479\" data-original-width=\"883\" height=\"174\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVG1ASsWq6o07U19DkHPeH14fILRFX2qe3jBear7Uja94xLS-_t0WB4tWT6HrsuA4zgyR5YHpFRJqHzO-F-4ff1-cwtlgleNoQYFimyRUQymhe3blkN9MfOKYo-lboKcgP38ZYwGpoyF0/w320-h174/segitiga03.png\" title=\"slider di GeoGebra\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"3-create-two-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-create-two-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Create Two Triangles\"\u003e\u003c/a\u003e3. Create Two Triangles\u003c/h3\u003e\u003cp\u003eConstruct a triangle $ABC$. Click *Polygon*, click the three point positions in the Graph View and return to the starting point, then we get the triangle $ABC$. Create another triangle that coincides with triangle $ABC$ in the same way as above. Look at the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeCJ5_5XekX70RzUPBmsWaE_lDHZchvLkywGZyyebIJKJ-AefRhpUy5hY2YQv7gW9BcrgExzSk3BFtrNAgwj7kfM2IdB1aQDQ3tU0T3XOkjKoHq5ISsS9y_gBGUJ7IZUfQZ2EttSKTWKY/s931/segitiga04.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"segitiga ABC dan segitiga BCD\" border=\"0\" data-original-height=\"458\" data-original-width=\"931\" height=\"157\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeCJ5_5XekX70RzUPBmsWaE_lDHZchvLkywGZyyebIJKJ-AefRhpUy5hY2YQv7gW9BcrgExzSk3BFtrNAgwj7kfM2IdB1aQDQ3tU0T3XOkjKoHq5ISsS9y_gBGUJ7IZUfQZ2EttSKTWKY/w320-h157/segitiga04.png\" title=\"segitiga ABC dan segitiga BCD\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"4-create-the-midpoint-of-side-ab\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-create-the-midpoint-of-side-ab\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Create the Midpoint of Side AB\"\u003e\u003c/a\u003e4. Create the Midpoint of Side AB\u003c/h3\u003e\u003cp\u003eCreate the midpoint of side $AB$. Click the Midpoint or Center, click point $A$ and point $B$, then point $E$ appears.\u003c/p\u003e\n\u003ch3 id=\"5-rotation-of-triangle-abc-about-the-midpoint\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-rotation-of-triangle-abc-about-the-midpoint\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Rotation of Triangle ABC about the Midpoint\"\u003e\u003c/a\u003e5. Rotation of Triangle ABC about the Midpoint\u003c/h3\u003e\u003cp\u003eTo make triangle $ABC$ rotate about point $E$ clockwise. Click *Rotation of an Object around a Point with an Angle*, click on triangle $ABC$, click point $E$, then a box appears. Write with $\\alpha$, select clockwise, click Ok. The result is the triangle $A'B'C'$.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFacMB9r0wkrOxWsoUptEPnffBy1F0PCfgrjYVAe7zNzjSHjqohJROw7yXsxDo3eGYrgH1dRP0xAv48k8lCL1d3TKN-NC5rJtsZBRHoM1DX6YCA0fru1ktO5AcTNyDBmyLZw6H2YZPYog/s919/segitiga05.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"rotasi segitiga ABC\" border=\"0\" data-original-height=\"469\" data-original-width=\"919\" height=\"163\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFacMB9r0wkrOxWsoUptEPnffBy1F0PCfgrjYVAe7zNzjSHjqohJROw7yXsxDo3eGYrgH1dRP0xAv48k8lCL1d3TKN-NC5rJtsZBRHoM1DX6YCA0fru1ktO5AcTNyDBmyLZw6H2YZPYog/w320-h163/segitiga05.png\" title=\"rotasi segitiga ABC\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"6-move-the-rotation-result-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-move-the-rotation-result-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Move the Rotation Result Triangle\"\u003e\u003c/a\u003e6. Move the Rotation Result Triangle\u003c/h3\u003e\u003cp\u003eMove triangle $A'B'C'$ using *glide*, so we get a quadrilateral.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8GC09EkHNrMHUq-nKjtdV8X7UbpD4YiK_TovIqZF1olpRlW6e4NXWLtsC7RGex-cwUvPc1LTQRekMHy8PgcyS13h3GJLOZqaLcwPIhl8U-UzNlpljAHZNYEAb5vTfI-S-x3V72Ur_EHk/s757/segitiga07.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"rotasi segitiga membentuk segiempat\" border=\"0\" data-original-height=\"504\" data-original-width=\"757\" height=\"213\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8GC09EkHNrMHUq-nKjtdV8X7UbpD4YiK_TovIqZF1olpRlW6e4NXWLtsC7RGex-cwUvPc1LTQRekMHy8PgcyS13h3GJLOZqaLcwPIhl8U-UzNlpljAHZNYEAb5vTfI-S-x3V72Ur_EHk/w320-h213/segitiga07.png\" title=\"rotasi segitiga ABC\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ-69StjjZCentXui9vQDY7QP09bRnjaLSx1nk_e8GwMvwEXaYVz5rjcV29qMLGsV8GjrRUQkvDyL4ZgYJSgzOc56qC1XN8nqTnpfONOw0v7wqG1w04HuNRWhhSjBIrqkO8blY-BZWvUE/s760/segitiga09.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"segitiga membentuk segiempat\" border=\"0\" data-original-height=\"447\" data-original-width=\"760\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ-69StjjZCentXui9vQDY7QP09bRnjaLSx1nk_e8GwMvwEXaYVz5rjcV29qMLGsV8GjrRUQkvDyL4ZgYJSgzOc56qC1XN8nqTnpfONOw0v7wqG1w04HuNRWhhSjBIrqkO8blY-BZWvUE/s320/segitiga09.png\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"7-do-the-same-for-triangle-bcd\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-do-the-same-for-triangle-bcd\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Do the same for Triangle BCD\"\u003e\u003c/a\u003e7. Do the same for Triangle BCD\u003c/h3\u003e\u003cp\u003eDo the same thing starting from step $4$ to step $6$ to make a rotation of triangle $BCD$ about the midpoint $BD$. Keep selecting the corner $\\alpha$, so that it continues to use the slide that was created together. Look at the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9oDEr408f38F9TkEhFYmzhV-peB1UOg_OY8YWpp_meH5aynb-SfLXWnqQzV157XLiD9J6OfifU3LQkb3h9kW6xT0k2fYPfXFRpVMyA4nGUGKXZ1CecszmYrdd4Ss1WHtLBdEglVcaj4E/s753/segitiga12.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"rotasi segitiga BCD\" border=\"0\" data-original-height=\"609\" data-original-width=\"753\" height=\"259\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9oDEr408f38F9TkEhFYmzhV-peB1UOg_OY8YWpp_meH5aynb-SfLXWnqQzV157XLiD9J6OfifU3LQkb3h9kW6xT0k2fYPfXFRpVMyA4nGUGKXZ1CecszmYrdd4Ss1WHtLBdEglVcaj4E/w320-h259/segitiga12.png\" title=\"rotasi segitiga BCD\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbY2OPmCixBeLPSYSvkg2NVNOS5q2GJkopK07iQgixJMJJR4NpspKj0kbQLi8brD4F-LwJE1DFgkJKneZ4rlgOpB9fwyMa_k1ZGqFXvTu1F170rcQiCVbSu-YWT3ECoH68jC1hPzZv0YE/s704/segitiga11.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"segi empat dari tiga segitiga\" border=\"0\" data-original-height=\"609\" data-original-width=\"704\" height=\"277\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbY2OPmCixBeLPSYSvkg2NVNOS5q2GJkopK07iQgixJMJJR4NpspKj0kbQLi8brD4F-LwJE1DFgkJKneZ4rlgOpB9fwyMa_k1ZGqFXvTu1F170rcQiCVbSu-YWT3ECoH68jC1hPzZv0YE/w320-h277/segitiga11.png\" title=\"segi empat dari tiga segitiga\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"conclusion-on-proving-the-area-of-a-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion-on-proving-the-area-of-a-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion on Proving the Area of ​​a Triangle\"\u003e\u003c/a\u003eConclusion on Proving the Area of ​​a Triangle\u003c/h3\u003e\u003cp\u003eIt can be seen that the area of ​​the triangle is equal to half the area of ​​the quadrilateral.\u003c/p\u003e\n\\(\\begin{aligned}\n\\text{Length of rectangle} \u0026= \\text{length of base of triangle}\\\\\n\\text{width of rectangle} \u0026= \\text{height of triangle}\\\\\n\\text{Area of the triangle} \u0026= \\frac{1}{2}\\text{of the length of the rectangle} \\times \\text{width of the rectangle}\\\\\n\u0026= \\frac{1}{2}\\times \\text{base of triangle} \\times \\text{height of triangle}\n\\end{aligned}\\)\n### 8. Enable Animation\n\u003cp\u003eSo that the image moves, then right click on the slide, click *Live Animation*.\u003c/p\u003e\n\u003cp\u003eFinally, trim the triangles, remove the axis lines, grids, and give the triangles and glides an attractive coloring.\u003c/p\u003e\n\u003ch3 id=\"the-final-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-final-result\" class=\"header-mark\" aria-label=\"Direct link to section: The final result\"\u003e\u003c/a\u003eThe final result\u003c/h3\u003e\u003cp\u003eBelow is an example of a triangle that has been created, click **play icon** at the bottom left, or slide the slider to see the triangle change.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/teappdfg/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; width: 100%; max-width: 650px;\" title=\"Luas Segitiga\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThis is a guide on how to prove the area of ​​a triangle using the area of ​​a quadrilateral using the GeoGebra application. If there are errors, please correct them, if you don't understand, let's discuss. If you have input, please write a comment here, if you like, I will make another guide 😂. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Inserting Images", "date_published" : "2025-12-14T06:31:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menyisipkan-gambar/", "url" : "https://www.aantriono.com/en/menyisipkan-gambar/", "summary": " Inserting Images in LaTeXImages are often needed to clarify explanations in articles, reports, worksheets, and scientific writing. In LaTeX\\LaTeXLATE​X, you can insert external image files and then control their size, position, rotation, and relationship to surrounding text.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"inserting-images-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#inserting-images-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Inserting Images in LaTeX\"\u003e\u003c/a\u003eInserting Images in LaTeX\u003c/h2\u003e\u003cp\u003eImages are often needed to clarify explanations in articles, reports, worksheets, and scientific writing. In \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, you can insert external image files and then control their size, position, rotation, and relationship to surrounding text.\u003c/p\u003e\n\u003ch3 id=\"basic-requirements\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-requirements\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Requirements\"\u003e\u003c/a\u003eBasic Requirements\u003c/h3\u003e\u003cp\u003eBefore inserting an image, keep these points in mind:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eadd the \u003ccode\u003egraphicx\u003c/code\u003e package in the preamble,\u003c/li\u003e\n\u003cli\u003estore the image file in the same folder as the document or use the correct path,\u003c/li\u003e\n\u003cli\u003euse \u003ccode\u003e\\caption{...}\u003c/code\u003e when the image needs a caption.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you are working in Overleaf, upload the image file into the same project first.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHhoFNwj0hn9VUP2H2ICPzu6qxpdgzbHpUajYMUqJv5lQ1j3INW5471G7Gq3-lhRr47xRQ51a3nFMqqx63ITrRTC5ZIlfor6lLbXm512Sd-up96kIlyZDd7_LNhXClNuQF-UaotAQbWLo/w320-h158/blog1.png\" title=\"Uploading image files in Overleaf\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHhoFNwj0hn9VUP2H2ICPzu6qxpdgzbHpUajYMUqJv5lQ1j3INW5471G7Gq3-lhRr47xRQ51a3nFMqqx63ITrRTC5ZIlfor6lLbXm512Sd-up96kIlyZDd7_LNhXClNuQF-UaotAQbWLo/w320-h158/blog1.png\" data-sub-html=\"\u003ch2\u003eUploading image files in Overleaf\u003c/h2\u003e\u003cp\u003eUploading image files in Overleaf\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHhoFNwj0hn9VUP2H2ICPzu6qxpdgzbHpUajYMUqJv5lQ1j3INW5471G7Gq3-lhRr47xRQ51a3nFMqqx63ITrRTC5ZIlfor6lLbXm512Sd-up96kIlyZDd7_LNhXClNuQF-UaotAQbWLo/w320-h158/blog1.png' alt=\"Uploading image files in Overleaf\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eUploading image files in Overleaf\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHhoFNwj0hn9VUP2H2ICPzu6qxpdgzbHpUajYMUqJv5lQ1j3INW5471G7Gq3-lhRr47xRQ51a3nFMqqx63ITrRTC5ZIlfor6lLbXm512Sd-up96kIlyZDd7_LNhXClNuQF-UaotAQbWLo/s1119/blog1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHhoFNwj0hn9VUP2H2ICPzu6qxpdgzbHpUajYMUqJv5lQ1j3INW5471G7Gq3-lhRr47xRQ51a3nFMqqx63ITrRTC5ZIlfor6lLbXm512Sd-up96kIlyZDd7_LNhXClNuQF-UaotAQbWLo/s1119/blog1.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"inserting-a-single-image\" class=\"headerLink\"\u003e\n \u003ca href=\"#inserting-a-single-image\" class=\"header-mark\" aria-label=\"Direct link to section: Inserting a Single Image\"\u003e\u003c/a\u003eInserting a Single Image\u003c/h3\u003e\u003cp\u003eThe simplest example looks like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.5\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eExample image\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifuX7XwDbRjKR7n3ErnrxzAMltKG76OJiGPzFoHymH4_zttCNldzuDXA3mWFYN8imyQDjVBYnw1y8spVzWcnWJ9CGbdnKlbA_oOxLnzNqN8o7k7xNLTFfRz6cCJLm2_kr4yfS06lyeCj8/w320-h99/gambar1.png\" title=\"Single image example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifuX7XwDbRjKR7n3ErnrxzAMltKG76OJiGPzFoHymH4_zttCNldzuDXA3mWFYN8imyQDjVBYnw1y8spVzWcnWJ9CGbdnKlbA_oOxLnzNqN8o7k7xNLTFfRz6cCJLm2_kr4yfS06lyeCj8/w320-h99/gambar1.png\" data-sub-html=\"\u003ch2\u003eSingle image example in LaTeX\u003c/h2\u003e\u003cp\u003eSingle image example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifuX7XwDbRjKR7n3ErnrxzAMltKG76OJiGPzFoHymH4_zttCNldzuDXA3mWFYN8imyQDjVBYnw1y8spVzWcnWJ9CGbdnKlbA_oOxLnzNqN8o7k7xNLTFfRz6cCJLm2_kr4yfS06lyeCj8/w320-h99/gambar1.png' alt=\"Single image example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSingle image example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifuX7XwDbRjKR7n3ErnrxzAMltKG76OJiGPzFoHymH4_zttCNldzuDXA3mWFYN8imyQDjVBYnw1y8spVzWcnWJ9CGbdnKlbA_oOxLnzNqN8o7k7xNLTFfRz6cCJLm2_kr4yfS06lyeCj8/s1198/gambar1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifuX7XwDbRjKR7n3ErnrxzAMltKG76OJiGPzFoHymH4_zttCNldzuDXA3mWFYN8imyQDjVBYnw1y8spVzWcnWJ9CGbdnKlbA_oOxLnzNqN8o7k7xNLTFfRz6cCJLm2_kr4yfS06lyeCj8/s1198/gambar1.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"inserting-two-or-more-images\" class=\"headerLink\"\u003e\n \u003ca href=\"#inserting-two-or-more-images\" class=\"header-mark\" aria-label=\"Direct link to section: Inserting Two or More Images\"\u003e\u003c/a\u003eInserting Two or More Images\u003c/h3\u003e\u003cp\u003eTo show multiple images in one section, you can place multiple \u003ccode\u003e\\includegraphics\u003c/code\u003e commands together.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 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21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.35\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage1.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.35\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage2.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eTwo images in one figure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb5h2Xz6S_79NmBkWJwAoW2cdba9-qDOyLhHvyV6yvvqjoUXGq6VUtyXdnQ0Cg47D_H-NmndYj66Q4xBbKxCRaJiNDSkrah3bhTheHvhyphenhyphen8Rn5nKZQJFw5tc0JPLyJo8MP50SQAqv3Girc/w320-h169/gambar2.png\" title=\"Two images example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb5h2Xz6S_79NmBkWJwAoW2cdba9-qDOyLhHvyV6yvvqjoUXGq6VUtyXdnQ0Cg47D_H-NmndYj66Q4xBbKxCRaJiNDSkrah3bhTheHvhyphenhyphen8Rn5nKZQJFw5tc0JPLyJo8MP50SQAqv3Girc/w320-h169/gambar2.png\" data-sub-html=\"\u003ch2\u003eTwo images example in LaTeX\u003c/h2\u003e\u003cp\u003eTwo images example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb5h2Xz6S_79NmBkWJwAoW2cdba9-qDOyLhHvyV6yvvqjoUXGq6VUtyXdnQ0Cg47D_H-NmndYj66Q4xBbKxCRaJiNDSkrah3bhTheHvhyphenhyphen8Rn5nKZQJFw5tc0JPLyJo8MP50SQAqv3Girc/w320-h169/gambar2.png' alt=\"Two images example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eTwo images example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb5h2Xz6S_79NmBkWJwAoW2cdba9-qDOyLhHvyV6yvvqjoUXGq6VUtyXdnQ0Cg47D_H-NmndYj66Q4xBbKxCRaJiNDSkrah3bhTheHvhyphenhyphen8Rn5nKZQJFw5tc0JPLyJo8MP50SQAqv3Girc/s984/gambar2.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb5h2Xz6S_79NmBkWJwAoW2cdba9-qDOyLhHvyV6yvvqjoUXGq6VUtyXdnQ0Cg47D_H-NmndYj66Q4xBbKxCRaJiNDSkrah3bhTheHvhyphenhyphen8Rn5nKZQJFw5tc0JPLyJo8MP50SQAqv3Girc/s984/gambar2.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"resizing-images\" class=\"headerLink\"\u003e\n \u003ca href=\"#resizing-images\" class=\"header-mark\" aria-label=\"Direct link to section: Resizing Images\"\u003e\u003c/a\u003eResizing Images\u003c/h3\u003e\u003cp\u003eCommon options include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\includegraphics[scale=...]{image}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\includegraphics[width=...,height=...]{image}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\includegraphics[width=...]{image}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\includegraphics[width=\\textwidth]{image}\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.6\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"rotating-images\" class=\"headerLink\"\u003e\n \u003ca href=\"#rotating-images\" class=\"header-mark\" aria-label=\"Direct link to section: Rotating Images\"\u003e\u003c/a\u003eRotating Images\u003c/h3\u003e\u003cp\u003eYou can rotate an image with the \u003ccode\u003eangle\u003c/code\u003e option.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.4\\textwidth,angle=45]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS3IwrbRw9ecjH33_uLfN_T6wqvWoN4zP5Ry5U_BLZliUzO9xCUszWz9STNhzSl3_oFyI5mKUnhYv74lkaKRcMLtbUy_Yst_HOoOUm63x7S50hXITkLY5PVX1YbeerFtqjeigCX2sQko0/w320-h152/gambar3.png\" title=\"Rotated image example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS3IwrbRw9ecjH33_uLfN_T6wqvWoN4zP5Ry5U_BLZliUzO9xCUszWz9STNhzSl3_oFyI5mKUnhYv74lkaKRcMLtbUy_Yst_HOoOUm63x7S50hXITkLY5PVX1YbeerFtqjeigCX2sQko0/w320-h152/gambar3.png\" data-sub-html=\"\u003ch2\u003eRotated image example in LaTeX\u003c/h2\u003e\u003cp\u003eRotated image example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS3IwrbRw9ecjH33_uLfN_T6wqvWoN4zP5Ry5U_BLZliUzO9xCUszWz9STNhzSl3_oFyI5mKUnhYv74lkaKRcMLtbUy_Yst_HOoOUm63x7S50hXITkLY5PVX1YbeerFtqjeigCX2sQko0/w320-h152/gambar3.png' alt=\"Rotated image example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eRotated image example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS3IwrbRw9ecjH33_uLfN_T6wqvWoN4zP5Ry5U_BLZliUzO9xCUszWz9STNhzSl3_oFyI5mKUnhYv74lkaKRcMLtbUy_Yst_HOoOUm63x7S50hXITkLY5PVX1YbeerFtqjeigCX2sQko0/s982/gambar3.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS3IwrbRw9ecjH33_uLfN_T6wqvWoN4zP5Ry5U_BLZliUzO9xCUszWz9STNhzSl3_oFyI5mKUnhYv74lkaKRcMLtbUy_Yst_HOoOUm63x7S50hXITkLY5PVX1YbeerFtqjeigCX2sQko0/s982/gambar3.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"image-alignment\" class=\"headerLink\"\u003e\n \u003ca href=\"#image-alignment\" class=\"header-mark\" aria-label=\"Direct link to section: Image Alignment\"\u003e\u003c/a\u003eImage Alignment\u003c/h3\u003e\u003cp\u003eTo control spacing between images, you can use horizontal and vertical spacing commands such as \u003ccode\u003e\\quad\u003c/code\u003e or \u003ccode\u003e\\baselineskip\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.3\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage1.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\quad\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.3\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage2.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQATYTkDEaoPSwj1zuHeGxJ2tZV_ZEsxETYohPRlLY9FDCWe2vn1u6Af_DS9q5SwCjP5o_O07w2MN7vCmx3FV3yAXM-Wdk7XW61GnsGTCcbKZEGrnbpZC7CebSeGoRd3OcQKMjVWPno1A/w320-h125/gambar4.png\" title=\"Image alignment example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQATYTkDEaoPSwj1zuHeGxJ2tZV_ZEsxETYohPRlLY9FDCWe2vn1u6Af_DS9q5SwCjP5o_O07w2MN7vCmx3FV3yAXM-Wdk7XW61GnsGTCcbKZEGrnbpZC7CebSeGoRd3OcQKMjVWPno1A/w320-h125/gambar4.png\" data-sub-html=\"\u003ch2\u003eImage alignment example in LaTeX\u003c/h2\u003e\u003cp\u003eImage alignment example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQATYTkDEaoPSwj1zuHeGxJ2tZV_ZEsxETYohPRlLY9FDCWe2vn1u6Af_DS9q5SwCjP5o_O07w2MN7vCmx3FV3yAXM-Wdk7XW61GnsGTCcbKZEGrnbpZC7CebSeGoRd3OcQKMjVWPno1A/w320-h125/gambar4.png' alt=\"Image alignment example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eImage alignment example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQATYTkDEaoPSwj1zuHeGxJ2tZV_ZEsxETYohPRlLY9FDCWe2vn1u6Af_DS9q5SwCjP5o_O07w2MN7vCmx3FV3yAXM-Wdk7XW61GnsGTCcbKZEGrnbpZC7CebSeGoRd3OcQKMjVWPno1A/s1212/gambar4.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQATYTkDEaoPSwj1zuHeGxJ2tZV_ZEsxETYohPRlLY9FDCWe2vn1u6Af_DS9q5SwCjP5o_O07w2MN7vCmx3FV3yAXM-Wdk7XW61GnsGTCcbKZEGrnbpZC7CebSeGoRd3OcQKMjVWPno1A/s1212/gambar4.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"image-position-in-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#image-position-in-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: Image Position in the Document\"\u003e\u003c/a\u003eImage Position in the Document\u003c/h3\u003e\u003cp\u003eImages are usually placed inside the \u003ccode\u003efigure\u003c/code\u003e environment.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 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4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=0.5\\textwidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eImage at a chosen position\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwCmh5Vkr_QrRLFRdvxR2JUNqmF4kaHEEiQeY5j5EvUVxRAazo1Q6Fl8V1FuS5BtjbfwBXVVAb7lGPrhaThqKYm97I4nl3Rc-xpu19L5UV1E04QSZW2MmaBtJ9In2J_Rp_6rd7P5ICbFA/w320-h98/gambar5.png\" title=\"Image position example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwCmh5Vkr_QrRLFRdvxR2JUNqmF4kaHEEiQeY5j5EvUVxRAazo1Q6Fl8V1FuS5BtjbfwBXVVAb7lGPrhaThqKYm97I4nl3Rc-xpu19L5UV1E04QSZW2MmaBtJ9In2J_Rp_6rd7P5ICbFA/w320-h98/gambar5.png\" data-sub-html=\"\u003ch2\u003eImage position example in LaTeX\u003c/h2\u003e\u003cp\u003eImage position example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwCmh5Vkr_QrRLFRdvxR2JUNqmF4kaHEEiQeY5j5EvUVxRAazo1Q6Fl8V1FuS5BtjbfwBXVVAb7lGPrhaThqKYm97I4nl3Rc-xpu19L5UV1E04QSZW2MmaBtJ9In2J_Rp_6rd7P5ICbFA/w320-h98/gambar5.png' alt=\"Image position example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eImage position example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwCmh5Vkr_QrRLFRdvxR2JUNqmF4kaHEEiQeY5j5EvUVxRAazo1Q6Fl8V1FuS5BtjbfwBXVVAb7lGPrhaThqKYm97I4nl3Rc-xpu19L5UV1E04QSZW2MmaBtJ9In2J_Rp_6rd7P5ICbFA/s1212/gambar5.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwCmh5Vkr_QrRLFRdvxR2JUNqmF4kaHEEiQeY5j5EvUVxRAazo1Q6Fl8V1FuS5BtjbfwBXVVAb7lGPrhaThqKYm97I4nl3Rc-xpu19L5UV1E04QSZW2MmaBtJ9In2J_Rp_6rd7P5ICbFA/s1212/gambar5.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eCommon placement options for \u003ccode\u003e\\begin{figure}[...]\u003c/code\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003eh\u003c/code\u003e: here\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003et\u003c/code\u003e: top\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eb\u003c/code\u003e: bottom\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003ep\u003c/code\u003e: special float page\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e!\u003c/code\u003e: force the placement more strongly\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eH\u003c/code\u003e: place it exactly here, usually with the \u003ccode\u003efloat\u003c/code\u003e package\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eFor alignment:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\centering\u003c/code\u003e centers the image,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\flushleft\u003c/code\u003e aligns it left,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\flushright\u003c/code\u003e aligns it right.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"multiple-images-with-subfigures\" class=\"headerLink\"\u003e\n \u003ca href=\"#multiple-images-with-subfigures\" class=\"header-mark\" aria-label=\"Direct link to section: Multiple Images with Subfigures\"\u003e\u003c/a\u003eMultiple Images with Subfigures\u003c/h3\u003e\u003cp\u003eIf readers need to compare several images, use subfigures or subcaptions.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n 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224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esubcaption\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esubfigure\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.45\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=\\linewidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage1.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eFirst image\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esubfigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esubfigure\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.45\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=\\linewidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage2.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSecond image\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esubfigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eTwo subfigures\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivIX6w-i5Dn3XfaLIA8l5aH1y8WPhn8vtefe9mjDYMF2mUN7zc9LeYWOh3bnG_Hf0vPUNoO3IlYaMBopXgoFrz3LjMv4XTL89FLMLbQPkSu9d9fReEKJU2HjijwY86jXJpDTmi2IKVfsM/w320-h120/gambar01.png\" title=\"Subfigure example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivIX6w-i5Dn3XfaLIA8l5aH1y8WPhn8vtefe9mjDYMF2mUN7zc9LeYWOh3bnG_Hf0vPUNoO3IlYaMBopXgoFrz3LjMv4XTL89FLMLbQPkSu9d9fReEKJU2HjijwY86jXJpDTmi2IKVfsM/w320-h120/gambar01.png\" data-sub-html=\"\u003ch2\u003eSubfigure example in LaTeX\u003c/h2\u003e\u003cp\u003eSubfigure example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivIX6w-i5Dn3XfaLIA8l5aH1y8WPhn8vtefe9mjDYMF2mUN7zc9LeYWOh3bnG_Hf0vPUNoO3IlYaMBopXgoFrz3LjMv4XTL89FLMLbQPkSu9d9fReEKJU2HjijwY86jXJpDTmi2IKVfsM/w320-h120/gambar01.png' alt=\"Subfigure example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSubfigure example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivIX6w-i5Dn3XfaLIA8l5aH1y8WPhn8vtefe9mjDYMF2mUN7zc9LeYWOh3bnG_Hf0vPUNoO3IlYaMBopXgoFrz3LjMv4XTL89FLMLbQPkSu9d9fReEKJU2HjijwY86jXJpDTmi2IKVfsM/s1223/gambar01.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivIX6w-i5Dn3XfaLIA8l5aH1y8WPhn8vtefe9mjDYMF2mUN7zc9LeYWOh3bnG_Hf0vPUNoO3IlYaMBopXgoFrz3LjMv4XTL89FLMLbQPkSu9d9fReEKJU2HjijwY86jXJpDTmi2IKVfsM/s1223/gambar01.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"image-and-text-with-minipage\" class=\"headerLink\"\u003e\n \u003ca href=\"#image-and-text-with-minipage\" class=\"header-mark\" aria-label=\"Direct link to section: Image and Text with Minipage\"\u003e\u003c/a\u003eImage and Text with Minipage\u003c/h3\u003e\u003cp\u003eTo place an image beside a block of text, \u003ccode\u003eminipage\u003c/code\u003e is a practical option.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.45\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=\\linewidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hfill\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.45\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Write the explanation here.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eminipage\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4VhHQKPchF6WtGQRDqNlwee0sGpYwdR8gEGiRdn7zHe7rTm7oohla1s3XttnsXCnHvDksBiUavpEtl04g_Q1aurgt20jff9in6x2GXeTF2LKPBp_sfk_r8wVUjoLYPd-NhvLsB-NEi1Y/w320-h131/gambar7.png\" title=\"Minipage example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4VhHQKPchF6WtGQRDqNlwee0sGpYwdR8gEGiRdn7zHe7rTm7oohla1s3XttnsXCnHvDksBiUavpEtl04g_Q1aurgt20jff9in6x2GXeTF2LKPBp_sfk_r8wVUjoLYPd-NhvLsB-NEi1Y/w320-h131/gambar7.png\" data-sub-html=\"\u003ch2\u003eMinipage example in LaTeX\u003c/h2\u003e\u003cp\u003eMinipage example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4VhHQKPchF6WtGQRDqNlwee0sGpYwdR8gEGiRdn7zHe7rTm7oohla1s3XttnsXCnHvDksBiUavpEtl04g_Q1aurgt20jff9in6x2GXeTF2LKPBp_sfk_r8wVUjoLYPd-NhvLsB-NEi1Y/w320-h131/gambar7.png' alt=\"Minipage example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eMinipage example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4VhHQKPchF6WtGQRDqNlwee0sGpYwdR8gEGiRdn7zHe7rTm7oohla1s3XttnsXCnHvDksBiUavpEtl04g_Q1aurgt20jff9in6x2GXeTF2LKPBp_sfk_r8wVUjoLYPd-NhvLsB-NEi1Y/s1233/gambar7.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4VhHQKPchF6WtGQRDqNlwee0sGpYwdR8gEGiRdn7zHe7rTm7oohla1s3XttnsXCnHvDksBiUavpEtl04g_Q1aurgt20jff9in6x2GXeTF2LKPBp_sfk_r8wVUjoLYPd-NhvLsB-NEi1Y/s1233/gambar7.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"alternative-wrapfig\" class=\"headerLink\"\u003e\n \u003ca href=\"#alternative-wrapfig\" class=\"header-mark\" aria-label=\"Direct link to section: Alternative: wrapfig\"\u003e\u003c/a\u003eAlternative: \u003ccode\u003ewrapfig\u003c/code\u003e\u003c/h3\u003e\u003cp\u003eIf you want text to wrap around an image, use the \u003ccode\u003ewrapfig\u003c/code\u003e package.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewrapfig\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewrapfigure\u003cspan class=\"nb\"\u003e}{\u003c/span\u003er\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.4\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\includegraphics\u003c/span\u003e\u003cspan class=\"na\"\u003e[width=\\linewidth]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eimage.png\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewrapfigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPap1kLQDcrVOy63X_CFfzUL6GdkJCQGpY8lsZJr9KXVS_oPDl1205f-Qk7ioQjj4VkgkK6Vbnnok3CSGkIaIWencKAxpjmudTssI7tAaHSDBwxQxd9OvwaJpL1XC1MKdu3YbKxfCXsM/w320-h147/gambar8.png\" title=\"Wrapfig example in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPap1kLQDcrVOy63X_CFfzUL6GdkJCQGpY8lsZJr9KXVS_oPDl1205f-Qk7ioQjj4VkgkK6Vbnnok3CSGkIaIWencKAxpjmudTssI7tAaHSDBwxQxd9OvwaJpL1XC1MKdu3YbKxfCXsM/w320-h147/gambar8.png\" data-sub-html=\"\u003ch2\u003eWrapfig example in LaTeX\u003c/h2\u003e\u003cp\u003eWrapfig example in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPap1kLQDcrVOy63X_CFfzUL6GdkJCQGpY8lsZJr9KXVS_oPDl1205f-Qk7ioQjj4VkgkK6Vbnnok3CSGkIaIWencKAxpjmudTssI7tAaHSDBwxQxd9OvwaJpL1XC1MKdu3YbKxfCXsM/w320-h147/gambar8.png' alt=\"Wrapfig example in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eWrapfig example in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPap1kLQDcrVOy63X_CFfzUL6GdkJCQGpY8lsZJr9KXVS_oPDl1205f-Qk7ioQjj4VkgkK6Vbnnok3CSGkIaIWencKAxpjmudTssI7tAaHSDBwxQxd9OvwaJpL1XC1MKdu3YbKxfCXsM/s1082/gambar8.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPap1kLQDcrVOy63X_CFfzUL6GdkJCQGpY8lsZJr9KXVS_oPDl1205f-Qk7ioQjj4VkgkK6Vbnnok3CSGkIaIWencKAxpjmudTssI7tAaHSDBwxQxd9OvwaJpL1XC1MKdu3YbKxfCXsM/s1082/gambar8.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThose are some of the most common ways to insert images in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. With \u003ccode\u003egraphicx\u003c/code\u003e and supporting environments such as \u003ccode\u003efigure\u003c/code\u003e, \u003ccode\u003esubfigure\u003c/code\u003e, \u003ccode\u003eminipage\u003c/code\u003e, and \u003ccode\u003ewrapfig\u003c/code\u003e, image layout becomes much more flexible and easier to manage.\u003c/p\u003e\n", "language": "en" }, { "title" : "Quadrilateral Rotation Animation", "date_published" : "2025-12-14T06:26:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/animasi-rotasi-segi-empat/", "url" : "https://www.aantriono.com/en/animasi-rotasi-segi-empat/", "summary": "This article explains how to create a rotation animation for a quadrilateral using GeoGebra.\nSteps to Create a Quadrilateral Rotation Animation in GeoGebra 1. Open GeoGebraOpen the GeoGebra application on your computer or device.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explains how to create a rotation animation for a quadrilateral using GeoGebra.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe0TXQ-QM_FxRpmTndJbaXngmSKhzIOk1RlRXmeaOrvoqECayT_k6rtBZPgTH45sDMl-938uteID5xVYTH-yz-2KnyBl-UlhU8hPTsDJ7kDp6HT0dxXIdGpflNY9o8caMuFhA9A8dM99Y/s564/poli.png' alt=\"Quadrilateral rotation\" \u003e\u003c/p\u003e\n\u003ch2 id=\"steps-to-create-a-quadrilateral-rotation-animation-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-a-quadrilateral-rotation-animation-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Create a Quadrilateral Rotation Animation in GeoGebra\"\u003e\u003c/a\u003eSteps to Create a Quadrilateral Rotation Animation in GeoGebra\u003c/h2\u003e\u003ch3 id=\"1-open-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open GeoGebra\"\u003e\u003c/a\u003e1. Open GeoGebra\u003c/h3\u003e\u003cp\u003eOpen the GeoGebra application on your computer or device.\u003c/p\u003e\n\u003ch3 id=\"2-create-a-quadrilateral-polygon\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-create-a-quadrilateral-polygon\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Create a Quadrilateral Polygon\"\u003e\u003c/a\u003e2. Create a Quadrilateral Polygon\u003c/h3\u003e\u003cp\u003eClick \u003cem\u003ePolygon\u003c/em\u003e, then click four points on the graph view and return to the starting point. The result will be a shape ABCD like the image below.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBcUnLV5bUKOHKbhX0G_C1kiEXvMrflSE8cPiiUWVniMil59QOAuniuvNRxlhQncAN2BgHHKM1IR9PMolHuGmrwRl5beZnISkDtSa5VszbXA2Fy29gMMdr8t9QiPxmDqvg5oEXhLkVuok/s525/poli1.png' alt=\"Quadrilateral in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch3 id=\"3-create-the-rotation-center\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-create-the-rotation-center\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Create the Rotation Center\"\u003e\u003c/a\u003e3. Create the Rotation Center\u003c/h3\u003e\u003cp\u003eCreate a point at the intersection of the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis. Click the \u003cem\u003ePoint\u003c/em\u003e tool, then click the point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. GeoGebra will create point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eE(0,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e as the center of rotation.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2xfrL6lNqCUFZeB5MBj6C-CVkGWlj63e3J7yEzBRRgBCWjjoINlJIGqY8oOffupiagRKx6-eOYWMJqr03W-EUU4M-yYmyeic7XMzVUTFASqsNpuYmKvjQco6Sq9TF5iABfJHURm7wNsw/s471/poli2.png' alt=\"Rotation center point\" \u003e\u003c/p\u003e\n\u003ch3 id=\"4-create-an-angle-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-create-an-angle-slider\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Create an Angle Slider\"\u003e\u003c/a\u003e4. Create an Angle Slider\u003c/h3\u003e\u003cp\u003eClick \u003cem\u003eSlider\u003c/em\u003e, then click anywhere on the graph view. In the dialog box, choose \u003cem\u003eAngle\u003c/em\u003e, rename \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003ccode\u003eangle\u003c/code\u003e, then click \u003cem\u003eOK\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFycj_KRdLASXuqX72chqDTVuF2NEsDufDeRfhLzF_IYdVJj0Bx4BEsXx0jGWN8jYsZRnxQt0v2FnjS62-I2UDKfRxuT0MHRlXv6NQU8ScXwgPQzLhpq4oeKqqs7KTiS5mQzheww3Scfk/s727/poli3.png' alt=\"Angle slider in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch3 id=\"5-rotate-the-polygon\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-rotate-the-polygon\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Rotate the Polygon\"\u003e\u003c/a\u003e5. Rotate the Polygon\u003c/h3\u003e\u003cp\u003eIn this example, the polygon is named \u003ccode\u003eq1\u003c/code\u003e. Type the following command in the input bar:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eRotate[q1,angle,E]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003ePress Enter, and the rotated image will appear as follows.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWBD8BlYRqxUFWCImCu75zOM_oB8iYwBSGz21beBuwILprtoVrdWAd6J3IRaANxMJRSdpsbGkrMjpy_S7syz-wTerlt8CVSXm4489kU2KDGYZ_07iUxEe0kW_WJvK6KNcYqh8Pt9weGLY/s681/poli4.png' alt=\"Quadrilateral rotation in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch3 id=\"6-enable-trace-on\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-enable-trace-on\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Enable Trace On\"\u003e\u003c/a\u003e6. Enable Trace On\u003c/h3\u003e\u003cp\u003eRight-click the rotated image, then choose \u003cem\u003eTrace On\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHFy0m_uEiGz-h_FEjwIUW64e3fOatueJhD6y5UUFTX9XlH2Zn0G5pDWHe1NRPXKQ41Qfvs1ZCy-1pvJO8Z6rX59d0V39nOqZYJ1oJloef9P9nLPWw8tyJRCgyBin5H0lcVuCjYeO2Mc8/s666/poli5.png' alt=\"Trace on in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch3 id=\"7-enable-animation-on\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-enable-animation-on\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Enable Animation On\"\u003e\u003c/a\u003e7. Enable Animation On\u003c/h3\u003e\u003cp\u003eTo make the shape rotate automatically, right-click the \u003cem\u003eslider\u003c/em\u003e and choose \u003cem\u003eAnimation On\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyGly_Jq2dR1TMAZzCZN_jYYA5u4S5eF368_3mOylJbqWOymE_0ExjAs6kMqOKeENwjM7vLGYNFGg0jezsNdS8KRANsJhgPCR2NtZwSfhuPsvE5kKrdx-PAr-1pBvLg5n58HMzF6xesOA/s718/poli6.png' alt=\"Animation on in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch3 id=\"8-add-variations-and-tidy-the-display\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-add-variations-and-tidy-the-display\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Add Variations and Tidy the Display\"\u003e\u003c/a\u003e8. Add Variations and Tidy the Display\u003c/h3\u003e\u003cp\u003eRepeat steps 2-7 to create a second quadrilateral. To make the animation more attractive:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003echange the color of the rotated shape in the algebra view or polygon settings;\u003c/li\u003e\n\u003cli\u003eremove labels from the created shapes;\u003c/li\u003e\n\u003cli\u003eadjust the slider color if needed.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"final-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#final-result\" class=\"header-mark\" aria-label=\"Direct link to section: Final Result\"\u003e\u003c/a\u003eFinal Result\u003c/h2\u003e\u003cp\u003eBelow is an example of the finished animation. Click the \u003cem\u003eplay\u003c/em\u003e icon in the lower-left corner, or move the slider manually.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/m74bqt6t/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; width: 100%; max-width: 650px;\" title=\"Rotasi segi empat\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThat concludes this guide to creating a quadrilateral rotation animation in GeoGebra. If you have suggestions or corrections, please leave them in the comments.\u003c/p\u003e\n", "language": "en" }, { "title" : "Cosecant Function Graph Animation", "date_published" : "2025-12-14T06:23:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/animasi-grafik-fungsi-cosecan/", "url" : "https://www.aantriono.com/en/animasi-grafik-fungsi-cosecan/", "summary": "Trigonometry comes from Greek which consists of the words \"trigonon\" which means triangle, and \"metron\" which means measure. According to its origin, trigonometry is a branch of science that tries to investigate the motion of celestial bodies such as the sun, moon and stars, including calculating/estimating their positions. Trigonometry is a comparison value defined in Cartesian coordinates or a right triangle. Trigonometry is identical to trigonometric functions which include sine (sin), cosine (cos), tangent (tan), cosecan (cosec), secan (sec), and cotangent (cotan) all of which are ways to determine the side of a triangle or the angle formed from two sides in a triangle.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eTrigonometry comes from Greek which consists of the words \"trigonon\" which means triangle, and \"metron\" which means measure. According to its origin, trigonometry is a branch of science that tries to investigate the motion of celestial bodies such as the sun, moon and stars, including calculating/estimating their positions. Trigonometry is a comparison value defined in Cartesian coordinates or a right triangle. Trigonometry is identical to trigonometric functions which include sine (sin), cosine (cos), tangent (tan), cosecan (cosec), secan (sec), and cotangent (cotan) all of which are ways to determine the side of a triangle or the angle formed from two sides in a triangle.\u003c/p\u003e\n\u003cp\u003eIn this article, I will create a guide on how to create animated graphs of the cosecan (cosec) function using GeoGebra. Look at the following graph of the cosecant function.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi92ntrDTTuDgBR3vbBgPMAY7wP4wt_T5Eb7U4hrmA8dMuONbdewSAJlLvS2PBFb4hOrEA6LBsn9-iN1amsERPMzcOCNvUd8fbzr14PbJ0zbCPvNgPGOpy_YoB9liXwiKZbbCkeXO5QEGw/s1161/cosec11.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"animasi grafik fungsi cosecan\" border=\"0\" data-original-height=\"584\" data-original-width=\"1161\" height=\"161\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi92ntrDTTuDgBR3vbBgPMAY7wP4wt_T5Eb7U4hrmA8dMuONbdewSAJlLvS2PBFb4hOrEA6LBsn9-iN1amsERPMzcOCNvUd8fbzr14PbJ0zbCPvNgPGOpy_YoB9liXwiKZbbCkeXO5QEGw/w320-h161/cosec11.png\" title=\"animasi grafik fungsi cosecan\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch2 id=\"steps-to-create-a-cosecant-function-graph-animation-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-a-cosecant-function-graph-animation-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Create a Cosecant Function Graph Animation in GeoGebra\"\u003e\u003c/a\u003eSteps to Create a Cosecant Function Graph Animation in GeoGebra\u003c/h2\u003e\u003cp\u003eThe steps are as follows:\u003c/p\u003e\n\u003ch3 id=\"1-open-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open GeoGebra\"\u003e\u003c/a\u003e1. Open GeoGebra\u003c/h3\u003e\u003cp\u003eOpen the GeoGebra application on your computer or device.\u003c/p\u003e\n\u003ch3 id=\"2-open-graphics-settings\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-open-graphics-settings\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Open Graphics Settings\"\u003e\u003c/a\u003e2. Open Graphics Settings\u003c/h3\u003e\u003cp\u003eRight click on the graphic area, then the Context-menu appears, click *Graphics*.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS2MwWQJqx9bOFCVxMpIXoCZrqpxss2GTruXjyLUprsd_pdibv3RB9A02Wfi4hGGx4UhWwKIcFCIJSj6hmp7eQixToCtkxpDoV169lG-D0h7PFTZNCtTtrbPHZ86vexbUeSvyMtU-u2ik/s719/cosec1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"tampilan grafik di GeoGebra\" border=\"0\" data-original-height=\"489\" data-original-width=\"719\" height=\"218\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS2MwWQJqx9bOFCVxMpIXoCZrqpxss2GTruXjyLUprsd_pdibv3RB9A02Wfi4hGGx4UhWwKIcFCIJSj6hmp7eQixToCtkxpDoV169lG-D0h7PFTZNCtTtrbPHZ86vexbUeSvyMtU-u2ik/w320-h218/cosec1.png\" title=\"tampilan grafik di GeoGebra\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"3-set-graphic-dimensions\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-set-graphic-dimensions\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Set Graphic Dimensions\"\u003e\u003c/a\u003e3. Set Graphic Dimensions\u003c/h3\u003e\u003cp\u003eThen fill in the tab *Basic* in part *Dimensions*: xMin = -20, xMax = 400 and yMin = -6, yMax = 6.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJDr1k2pCNuHSxq6jraU-7EgD59df8pjXX6Fm8yKJ4MEC6-P16vbz-gWSiLqrljov8Tv9fDReFf6TLe99mUbxguMa0imXjSxlEsKcsIPey0hRVRtO8VrF_dx_L_cs5s1NJBzWYq7Jz2m4/s588/cosec2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"tampilan menu basic di GeoGebra\" border=\"0\" data-original-height=\"588\" data-original-width=\"395\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJDr1k2pCNuHSxq6jraU-7EgD59df8pjXX6Fm8yKJ4MEC6-P16vbz-gWSiLqrljov8Tv9fDReFf6TLe99mUbxguMa0imXjSxlEsKcsIPey0hRVRtO8VrF_dx_L_cs5s1NJBzWYq7Jz2m4/w215-h320/cosec2.png\" title=\"tampilan menu basic di GeoGebra\" width=\"215\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"4-set-the-x-axis\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-set-the-x-axis\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Set the X Axis\"\u003e\u003c/a\u003e4. Set the X Axis\u003c/h3\u003e\u003cp\u003eFrom tab *xAxis* in part *Distance* type $30$. In the Units section select $^0$ (degrees). Click *OK*.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPuEJz2RQVk_9kuz7Pn7I0ZZ3rxfIw5gBSYBophyxv5x8xzsbqFxkW8-i9XqIuS2pDaNuTeAD_fO8KQDF7lvIlWbqcOxW4nvNgPhyphenhyphenGHEJoEunubioCqKwwwhdJgnvpN1VVd2K3Dhpqs1g/s435/cosec3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"tampilan bagian xAxis di GeoGebra\" border=\"0\" data-original-height=\"435\" data-original-width=\"389\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPuEJz2RQVk_9kuz7Pn7I0ZZ3rxfIw5gBSYBophyxv5x8xzsbqFxkW8-i9XqIuS2pDaNuTeAD_fO8KQDF7lvIlWbqcOxW4nvNgPhyphenhyphenGHEJoEunubioCqKwwwhdJgnvpN1VVd2K3Dhpqs1g/w286-h320/cosec3.png\" title=\"tampilan bagian xAxis di GeoGebra\" width=\"286\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"5-create-a-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-create-a-slider\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Create a Slider\"\u003e\u003c/a\u003e5. Create a Slider\u003c/h3\u003e\u003cp\u003eNext we create the slide. Click on tools *Sliders*, click on the graph area, then fill in the slider box. Name: a, Type: Number, Interval Min = $-5$, Max = $5$, Increment = $0.1$.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-4dv4uBrc3MO5LMb-v2NJTLPjfrRE0HUY-QFA-qyFyeObbQgV3RAmN9yNOyTA1H8U_awACa2iJCgJf3twjLFLYooRJjqW15QGjlQtc-O4aKip0TSSnCcjYM-gaCnpTVWxgWNB-8LCwrc/s837/cosec4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"slider di GeoGebra\" border=\"0\" data-original-height=\"426\" data-original-width=\"837\" height=\"163\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-4dv4uBrc3MO5LMb-v2NJTLPjfrRE0HUY-QFA-qyFyeObbQgV3RAmN9yNOyTA1H8U_awACa2iJCgJf3twjLFLYooRJjqW15QGjlQtc-O4aKip0TSSnCcjYM-gaCnpTVWxgWNB-8LCwrc/w320-h163/cosec4.png\" title=\"slider di GeoGebra\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"6-cosecan-function-input\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-cosecan-function-input\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Cosecan Function Input\"\u003e\u003c/a\u003e6. Cosecan Function Input\u003c/h3\u003e\u003cp\u003eThen on *Inputs*, type the function formula to be drawn: $f(x)=cosec(x^0)$ and $g(x)=a* cosec(x^0)$.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRSxoCIHaPjR5rHSA7kdDIYUqn_tBltn31EBgYQ59RK0l4Fh1ixn_C67nNRAXtygJ069XZiDpd0JuqxFkn-aCKLMZqgBWZhHnMlx5MEG28eOn_iGlEHfIJtS3IYsKu8fc__ErNc1obX98/s1356/cosec5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"grafik fungsi cosec\" border=\"0\" data-original-height=\"687\" data-original-width=\"1356\" height=\"162\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRSxoCIHaPjR5rHSA7kdDIYUqn_tBltn31EBgYQ59RK0l4Fh1ixn_C67nNRAXtygJ069XZiDpd0JuqxFkn-aCKLMZqgBWZhHnMlx5MEG28eOn_iGlEHfIJtS3IYsKu8fc__ErNc1obX98/w320-h162/cosec5.png\" title=\"grafik fungsi cosec\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003cp\u003eNext, a graph of the function $g(x)$ will appear:\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi600J9daGXVtYtlaQ3S3DY8teRRsiDyjIxiMRgc6miFMy4b_N6jqjLeyC_82LyvF13s6DnNiaxfwXk5mXJho6Nfy6zEae8c6541afq10bXD9HwWiwTVZjX703wXXdS8yCZCcOpjlozUXQ/s1351/cosec6.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"grafik fungsi g(x)\" border=\"0\" data-original-height=\"686\" data-original-width=\"1351\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi600J9daGXVtYtlaQ3S3DY8teRRsiDyjIxiMRgc6miFMy4b_N6jqjLeyC_82LyvF13s6DnNiaxfwXk5mXJho6Nfy6zEae8c6541afq10bXD9HwWiwTVZjX703wXXdS8yCZCcOpjlozUXQ/s320/cosec6.png\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"7-test-by-moving-the-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-test-by-moving-the-slider\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Test by Moving the Slider\"\u003e\u003c/a\u003e7. Test by Moving the Slider\u003c/h3\u003e\u003cp\u003eShift *sliders* left/right, pay attention to the changes that occur.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJxt2vMJgY-XP4ZVUmJ-CpkePeG0JEkBgm9u4cShGdhntslWHormjPctn-o3nXCnyC9I_kW9EI1Rv-6iX0Z3oXNF32ypeyrGH4s-QZrmqtJg4CcKjaZM-oJRuwUtoX_46v8zP_8VsjDz0/s1359/cosec7.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"grafik fungsi cosec dengan slider berbeda\" border=\"0\" data-original-height=\"648\" data-original-width=\"1359\" height=\"153\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJxt2vMJgY-XP4ZVUmJ-CpkePeG0JEkBgm9u4cShGdhntslWHormjPctn-o3nXCnyC9I_kW9EI1Rv-6iX0Z3oXNF32ypeyrGH4s-QZrmqtJg4CcKjaZM-oJRuwUtoX_46v8zP_8VsjDz0/w320-h153/cosec7.png\" title=\"grafik fungsi cosec\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"8-enable-animation-and-trace\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-enable-animation-and-trace\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Enable Animation and Trace\"\u003e\u003c/a\u003e8. Enable Animation and Trace\u003c/h3\u003e\u003cp\u003eAnimate *sliders*. Right-click on *sliders* so that the Context-menu appears, select it *Animation On*. Right click on the function $g(x)$, select it *Trace On*.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9OGrQp1mmSabUM1XNmRLHTdDCjlBAFgMJhRgxdtVeQNxUlPHWWvC_xIXpc4VaXfG5h686pFbtLzlwtf9-gGPs6tk_foE7IsMHRbumc9ts-9kuKZNx43MBmAt2LXz62v3VZHMpyDUo_XQ/s561/cosec9.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"animation on di slider\" border=\"0\" data-original-height=\"366\" data-original-width=\"561\" height=\"209\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9OGrQp1mmSabUM1XNmRLHTdDCjlBAFgMJhRgxdtVeQNxUlPHWWvC_xIXpc4VaXfG5h686pFbtLzlwtf9-gGPs6tk_foE7IsMHRbumc9ts-9kuKZNx43MBmAt2LXz62v3VZHMpyDUo_XQ/w320-h209/cosec9.png\" title=\"animation on di slider\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003cp\u003ePay attention to the changes that occur in the graph. It looks like the following picture.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifNBGUaRWoH9q0MhmbgjL10l4eOgnReg7Eo9O4p4lqDhnMquHhncnVLlCDrXsb0CzisYba_5_9Ap5rjM0Zhmxr1LaUDPbu7KWpUqy8vAqh24YiAupGdYnAQBeI4Qz1003ygv7ikNXzja4/s1360/cosec10.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"hasil animasi grafik fungsi cosec\" border=\"0\" data-original-height=\"660\" data-original-width=\"1360\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifNBGUaRWoH9q0MhmbgjL10l4eOgnReg7Eo9O4p4lqDhnMquHhncnVLlCDrXsb0CzisYba_5_9Ap5rjM0Zhmxr1LaUDPbu7KWpUqy8vAqh24YiAupGdYnAQBeI4Qz1003ygv7ikNXzja4/s320/cosec10.png\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"9-tidy-up-the-appearance-and-final-results\" class=\"headerLink\"\u003e\n \u003ca href=\"#9-tidy-up-the-appearance-and-final-results\" class=\"header-mark\" aria-label=\"Direct link to section: 9. Tidy up the appearance and final results\"\u003e\u003c/a\u003e9. Tidy up the appearance and final results\u003c/h3\u003e\u003cp\u003eNext, tidy up the display of graphics, sliders, and so on. Below is an example of an animated graph of the cosecant function. Click the icon **play** at the bottom left or move the slider.\u003c/p\u003e\n\u003ciframe scrolling=\"no\" title=\"Grafik Cosec\" src=\"https://www.geogebra.org/material/iframe/id/jhk5qyxe/width/650/height/276/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border:0px; width:100%; max-width:650px; height:276px;\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThis concludes the guide to creating an animated graph of the cosecant function in GeoGebra. If you have suggestions or corrections, please leave them in the comments. I hope this tutorial is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Circle Dilation Animation", "date_published" : "2025-12-14T06:19:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/animasi-dilatasi-lingkaran/", "url" : "https://www.aantriono.com/en/animasi-dilatasi-lingkaran/", "summary": "In grade IX transformation geometry, students learn reflection, rotation, translation, and dilation. This article explains how to create a circle dilation animation in GeoGebra.\nDilation is a transformation that changes the size of a figure, either enlarging or reducing it, without changing its shape.\nSteps to Create a Circle Dilation Animation in GeoGebra 1. Open GeoGebraOpen the GeoGebra application on your computer or device.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eIn grade IX transformation geometry, students learn reflection, rotation, translation, and dilation. This article explains how to create a circle dilation animation in GeoGebra.\u003c/p\u003e\n\u003cp\u003eDilation is a transformation that changes the size of a figure, either enlarging or reducing it, without changing its shape.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUgpm5RA6yA9kuoePhMfn24xxaPNWSllSa5f1zhJEXTcQyuN0rTlRg6Y_m-Mmj87jUK4RMIEXVi0cB6BaG75aAHBLbAKV1kJhyphenhyphenv7GiYaYmpzEPwzngzP7VIdGfaK4VIMVCKkwvzLgSXVg/s417/dilatasi1.png' alt=\"Circle dilation in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch2 id=\"steps-to-create-a-circle-dilation-animation-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-a-circle-dilation-animation-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Create a Circle Dilation Animation in GeoGebra\"\u003e\u003c/a\u003eSteps to Create a Circle Dilation Animation in GeoGebra\u003c/h2\u003e\u003ch3 id=\"1-open-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open GeoGebra\"\u003e\u003c/a\u003e1. Open GeoGebra\u003c/h3\u003e\u003cp\u003eOpen the GeoGebra application on your computer or device.\u003c/p\u003e\n\u003ch3 id=\"2-create-points-a-and-b\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-create-points-a-and-b\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Create Points A and B\"\u003e\u003c/a\u003e2. Create Points A and B\u003c/h3\u003e\u003cp\u003eCreate point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e at coordinates \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(3,2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e at coordinates \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE0tEC2pKnh1Tq8zr8oURIbTskoPVLXCYHb-gKiVRZqHLdaRiCtVUJpvvBHl6PD_1fClzs6D2Dwilqp8SlqXZ5PkKPqiB6ZVU1b95DyQYwp4tJ-qjYjkqATKUq15-v49hhDLT4eEElITI/s619/dilatasi2.png' alt=\"Creating points in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch3 id=\"3-create-a-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-create-a-slider\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Create a Slider\"\u003e\u003c/a\u003e3. Create a Slider\u003c/h3\u003e\u003cp\u003eClick the \u003cem\u003eSlider\u003c/em\u003e tool, then click anywhere on the workspace. Fill in the slider form as needed, similar to the example below.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh8hj5-tkP0SkjyOIFy81_Ka-GHP5sr5OztfWMTq8Fwgcdf9hh4yL9PVcG2B0e4uRFsgxnuXQtkspoQDTjO8mhtx6Ew_Ch2W5yWmaXgAW6Eu1asyLJXNRA1M1Xc1qWHtHElVBbMXamTBU/s885/dilatasi3.png' alt=\"Slider in GeoGebra\" \u003e\u003c/p\u003e\n\u003ch3 id=\"4-create-the-first-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-create-the-first-circle\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Create the First Circle\"\u003e\u003c/a\u003e4. Create the First Circle\u003c/h3\u003e\u003cp\u003eEnter the following command to display a circle centered at point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ec = Circle[A,2]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result will look like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5n1U6m3uGk22wFU2d-oO2MhRLnjNxfgVtfDa8wLwH3qZbfSNd8NnuInrPogeHCvtRP_KpHs3d7hYuIg-lW0MotMQJOfaaW3TDgf6mFjEnbygfRiChK4pb9sIou9fTutoixNVhc1mkUHw/s725/dilatasi4.png' alt=\"Circle centered at point A\" \u003e\u003c/p\u003e\n\u003ch3 id=\"5-create-point-c-and-the-second-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-create-point-c-and-the-second-circle\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Create Point C and the Second Circle\"\u003e\u003c/a\u003e5. Create Point C and the Second Circle\u003c/h3\u003e\u003cp\u003eCreate another point, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC(3,-2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then create a circle centered at point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with the command:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ee = Circle[C,2]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"6-create-the-dilated-points\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-create-the-dilated-points\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Create the Dilated Points\"\u003e\u003c/a\u003e6. Create the Dilated Points\u003c/h3\u003e\u003cp\u003eCreate the dilation images of points \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e using the following commands:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eA\u0026#39; = Dilate[A,a,B]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eC\u0026#39; = Dilate[C,a,B]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"7-create-the-dilated-circles\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-create-the-dilated-circles\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Create the Dilated Circles\"\u003e\u003c/a\u003e7. Create the Dilated Circles\u003c/h3\u003e\u003cp\u003eEnter the following commands to display the dilated circles:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ed = Dilate[Circle[A,2],a,B]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ef = Dilate[Circle[C,2],a,B]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result will look like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh02nQvxzeAt0P8UzkLMDdgetlnWTihY-MweugEH7rxQKgD6zMMbUu0nwamsKNVImgvOg2yIysYlDeVgpyCXvvl6YYjefHfvxv6cH3q6-05xQVSB5woWkk7mT6D5GB20w2kxeGvT5HPjLo/s909/dilatasi7.png' alt=\"Dilation of two circles\" \u003e\u003c/p\u003e\n\u003ch3 id=\"8-enable-trace\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-enable-trace\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Enable Trace\"\u003e\u003c/a\u003e8. Enable Trace\u003c/h3\u003e\u003cp\u003eTo make the movement easier to observe, right-click each circle, choose \u003cem\u003eObject Properties\u003c/em\u003e, then enable \u003cem\u003eShow Trace\u003c/em\u003e in the \u003cem\u003eBasic\u003c/em\u003e section.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgimODolHJpPahtgs84zbmpVX4fTICCvy9uVYn2v-WKsg_R6JtQ02xOPjo88HRokLkOSfR6dtIAiJuNSVhpxbu2FJ1rnbEnzo8Ec7lvpL0oQFNBTMmQpV9DFjLV40AM5lbnWHrCc2korL4/s599/dilatasi8.png' alt=\"GeoGebra object properties\" \u003e\u003c/p\u003e\n\u003ch3 id=\"9-change-the-color-and-move-the-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#9-change-the-color-and-move-the-slider\" class=\"header-mark\" aria-label=\"Direct link to section: 9. Change the Color and Move the Slider\"\u003e\u003c/a\u003e9. Change the Color and Move the Slider\u003c/h3\u003e\u003cp\u003eChange the color of one circle so the two circles are easier to distinguish. Then move the \u003cem\u003eslider\u003c/em\u003e to observe the enlargement and reduction process.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfgDLZT4PGqUV_CSyccYu8ayDariqev4v6sGr3Vm2Cbv_97Tio6BbU80Gbs9aoxp74RWvCmCKmZzWR5Ow71M0xZG5b59DlPYsitqPymasTbPYSX0R5nmzc0Wdf71S13jeIe9NVd6dwUR8/s1060/dilatasi9.png' alt=\"Circle size changes\" \u003e\u003c/p\u003e\n\u003ch3 id=\"10-tidy-the-appearance\" class=\"headerLink\"\u003e\n \u003ca href=\"#10-tidy-the-appearance\" class=\"header-mark\" aria-label=\"Direct link to section: 10. Tidy the Appearance\"\u003e\u003c/a\u003e10. Tidy the Appearance\u003c/h3\u003e\u003cp\u003eTidy the circles, slider, and graph display. Below is an example of the finished animation. Click the \u003cem\u003eplay\u003c/em\u003e icon in the lower-left corner or move the slider manually.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/kgmnxwvf/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; width: 100%; max-width: 650px;\" title=\"Dilatasi\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThat concludes this guide to creating a circle dilation animation in GeoGebra. If you have suggestions or corrections, please leave them in the comments.\u003c/p\u003e\n", "language": "en" }, { "title" : "Sine and Cosine Wave Curves", "date_published" : "2025-12-14T06:15:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/gelombang-kurva-sinus-dan-cosinus/", "url" : "https://www.aantriono.com/en/gelombang-kurva-sinus-dan-cosinus/", "summary": "Geogebra is a dynamic program that has facilities for visualizing or demonstrating mathematical concepts as well as a tool for constructing mathematical concepts. In this regard, on this occasion, I will provide guidance on creating and animating trigonometric function curves, namely sine curves and cosine curves. Look at the following image.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eGeogebra is a dynamic program that has facilities for visualizing or demonstrating mathematical concepts as well as a tool for constructing mathematical concepts. In this regard, on this occasion, I will provide guidance on creating and animating trigonometric function curves, namely sine curves and cosine curves. Look at the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQTjUeOrHEUhFDsPvfy0DvxcrAx0a-GQKxTgm-jUqTE7fTcZ5_UOpwAUiWSJFzVlzHtFRb3oL8fdSoIVGaCWeTMhZFvsqp_e3a_pK5fbIT-VlvE0wURhJJv29kUU9MCe4XXF-z9U2Vick/s615/kurvabergerak1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"kurva sinus dan cosinus\" border=\"0\" data-original-height=\"422\" data-original-width=\"615\" height=\"220\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQTjUeOrHEUhFDsPvfy0DvxcrAx0a-GQKxTgm-jUqTE7fTcZ5_UOpwAUiWSJFzVlzHtFRb3oL8fdSoIVGaCWeTMhZFvsqp_e3a_pK5fbIT-VlvE0wURhJJv29kUU9MCe4XXF-z9U2Vick/w320-h220/kurvabergerak1.png\" title=\"kurva sinus dan cosinus\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch2 id=\"steps-to-create-and-animate-sine-and-cosine-curves-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-and-animate-sine-and-cosine-curves-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Create and Animate Sine and Cosine Curves in GeoGebra\"\u003e\u003c/a\u003eSteps to Create and Animate Sine and Cosine Curves in GeoGebra\u003c/h2\u003e\u003cp\u003eThe way to create and animate trigonometric function curves, namely sine curves and cosine curves, is as follows:\u003c/p\u003e\n\u003ch3 id=\"1-open-the-geogebra-application\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-the-geogebra-application\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open the GeoGebra Application\"\u003e\u003c/a\u003e1. Open the GeoGebra Application\u003c/h3\u003e\u003cp\u003eOpen the GeoGebra application on your computer or device.\u003c/p\u003e\n\u003ch3 id=\"2-set-graphic-display\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-set-graphic-display\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Set Graphic Display\"\u003e\u003c/a\u003e2. Set Graphic Display\u003c/h3\u003e\u003cp\u003eRight click on Graph View. Look at the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx5TeTOmpYjJ7_G1NrqW1GB4VXe2M5o8gKHuT5PLR6lbVR-DCS2L9qoUYkh2BU-flq2JfUG35s7R-uW23qkHgyODMnC8-bLiL9AzFV2GzIyqklwPgb279WhcYkt7CJia7YxFRVrmb5qeU/s614/kurva+berg1erak.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"tampilan grafik di GeoGebra\" border=\"0\" data-original-height=\"507\" data-original-width=\"614\" height=\"264\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx5TeTOmpYjJ7_G1NrqW1GB4VXe2M5o8gKHuT5PLR6lbVR-DCS2L9qoUYkh2BU-flq2JfUG35s7R-uW23qkHgyODMnC8-bLiL9AzFV2GzIyqklwPgb279WhcYkt7CJia7YxFRVrmb5qeU/w320-h264/kurva+berg1erak.png\" title=\"tampilan grafik di GeoGebra\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003cp\u003eFill in the graph's abscissa and ordinate format as follows: X axis: min = $-10^0$ and max = $360^0$ Y axis: min = $- 2$ and max = $2$. Look at the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6V26qa0i5Tuj36HuOodq4Z8utsO0FfxnpO6Ytjk50Vf8Nus8vB0j-9WmhQY5_nmIDlRkSQkKqQQePpR1iR34b_Wp3VccM11m6eZ6BUCwNzbtXGifdKEgGt-HAAbqvclDg2uMDyuIQiOU/s598/kurva+bergerak2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"absis pada GeoGebra\" border=\"0\" data-original-height=\"598\" data-original-width=\"402\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6V26qa0i5Tuj36HuOodq4Z8utsO0FfxnpO6Ytjk50Vf8Nus8vB0j-9WmhQY5_nmIDlRkSQkKqQQePpR1iR34b_Wp3VccM11m6eZ6BUCwNzbtXGifdKEgGt-HAAbqvclDg2uMDyuIQiOU/w215-h320/kurva+bergerak2.png\" title=\"absis pada GeoGebra\" width=\"215\" /\u003e\n\u003c/a\u003e\n\u003cp\u003eIn the xAxis section, fill in according to the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2XVUyz1OiDBkxjcvsUu-f8aY34eDOlcEf3rquhOpNuQKPTLQ4OBZCL5zNcZrpGiW0iwApcZ33tTOtYpW2ToXtm_FUrtjbCqu1j3XYDe1q6rpJ4qmf5DpEXUDsGTClZyLso32kA7TTUXY/s413/kurva+bergerak3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"tampilan sumbu X di GeoGebra\" border=\"0\" data-original-height=\"413\" data-original-width=\"402\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2XVUyz1OiDBkxjcvsUu-f8aY34eDOlcEf3rquhOpNuQKPTLQ4OBZCL5zNcZrpGiW0iwApcZ33tTOtYpW2ToXtm_FUrtjbCqu1j3XYDe1q6rpJ4qmf5DpEXUDsGTClZyLso32kA7TTUXY/w311-h320/kurva+bergerak3.png\" title=\"tampilan sumbu X di GeoGebra\" width=\"311\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"3-create-a-sliderlaunch\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-create-a-sliderlaunch\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Create a Slider/Launch\"\u003e\u003c/a\u003e3. Create a Slider/Launch\u003c/h3\u003e\u003cp\u003eMake two *slider/glide* that is *slider/glide a* And *slider/glide b* with criteria as in the following image. How to click *Sliders* then click on Graphic Display, a box will appear, fill it in with a name *a* along with other data as shown in the image below.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBECqkEfc5u3eowB7VG1uA3-HShZGyblWfYG5bfGHbwS4YYOT9XL2hIHRp7gkdXEBeUZHxrPeKJljuLI8WK5yOXi0xG0kWj7LbvQ7vWr6bzkgdlYRmbaa_Wmrg8nDHjxv3W1N8A0cTP-o/s957/kurva+bergerak4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"tampilan luncuran/slider di GeoGebra\" border=\"0\" data-original-height=\"508\" data-original-width=\"957\" height=\"170\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBECqkEfc5u3eowB7VG1uA3-HShZGyblWfYG5bfGHbwS4YYOT9XL2hIHRp7gkdXEBeUZHxrPeKJljuLI8WK5yOXi0xG0kWj7LbvQ7vWr6bzkgdlYRmbaa_Wmrg8nDHjxv3W1N8A0cTP-o/w320-h170/kurva+bergerak4.png\" title=\"tampilan luncuran/slider di GeoGebra\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003cp\u003eClick again *Sliders* then click on Graphic Display, a box will appear, fill it in with a name *b* along with other data as below.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgM4_KCKcxSxl7uSttZtbN8kW4KB65u953VG8hJjzwSV1uB1wkzMyJnlQXvpKj81Cbb6R0qlarAbG_oT0b5VKACMRplhWNve6R9i57Vmd1TrxkirrS_0Xgf3S7TEePadpPFvdg1Gbga7I4/s956/kurva+bergerak5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"tampilan slider di GeoGebra\" border=\"0\" data-original-height=\"506\" data-original-width=\"956\" height=\"169\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgM4_KCKcxSxl7uSttZtbN8kW4KB65u953VG8hJjzwSV1uB1wkzMyJnlQXvpKj81Cbb6R0qlarAbG_oT0b5VKACMRplhWNve6R9i57Vmd1TrxkirrS_0Xgf3S7TEePadpPFvdg1Gbga7I4/w320-h169/kurva+bergerak5.png\" title=\"tampilan slider di GeoGebra\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003cp\u003eSo there are two results *slider/glide*.\u003c/p\u003e\n\u003ch3 id=\"4-create-sine-and-cosine-curves\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-create-sine-and-cosine-curves\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Create Sine and Cosine Curves\"\u003e\u003c/a\u003e4. Create Sine and Cosine Curves\u003c/h3\u003e\u003cp\u003eOn the box *Inputs* write the formula $y = a*sin(b*x^0)$\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFOtujNQAcbdIH8ttTbBU0XEQHObXuCNNMhTMilb2FM4FH9cqinUGr87CJvUPfaCwBIe4onXe43m4pwbFQiX2SgrplXH7YYfyGhZr9SQw5AzAw_EqxdiHlUaIeePD3Teo8wT4m_roG1M0/s1355/kurva+bergerak6.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"kurva fungsi sinus\" border=\"0\" data-original-height=\"703\" data-original-width=\"1355\" height=\"166\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFOtujNQAcbdIH8ttTbBU0XEQHObXuCNNMhTMilb2FM4FH9cqinUGr87CJvUPfaCwBIe4onXe43m4pwbFQiX2SgrplXH7YYfyGhZr9SQw5AzAw_EqxdiHlUaIeePD3Teo8wT4m_roG1M0/w320-h166/kurva+bergerak6.png\" title=\"kurva fungsi sinus\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003cp\u003eNext, type $y = a*cos(b*x^0)$ in the box *Inputs*\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic5Lvj74Lm81NTz9msyxhk6H2sERnZr77xcqykcP9xLPNnMH1JEIToHEam8W4R89ruYOj-Ygm-gBGj69rcH1Eb10O_hFi1J_kpwn6iTf9v-lY_CHlrMsZylm88PBqzX5SSTGq3ChhVEMI/s1349/kurva+bergerak7.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"kurva cosinus di GeoGebra\" border=\"0\" data-original-height=\"705\" data-original-width=\"1349\" height=\"167\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic5Lvj74Lm81NTz9msyxhk6H2sERnZr77xcqykcP9xLPNnMH1JEIToHEam8W4R89ruYOj-Ygm-gBGj69rcH1Eb10O_hFi1J_kpwn6iTf9v-lY_CHlrMsZylm88PBqzX5SSTGq3ChhVEMI/w320-h167/kurva+bergerak7.png\" title=\"kurva cosinus di GeoGebra\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"5-move-the-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-move-the-slider\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Move the Slider\"\u003e\u003c/a\u003e5. Move the Slider\u003c/h3\u003e\u003cp\u003eMove it *slider/glide a* And *slider/glide b*, then the graph curve will move.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_131sp21eN436FAPWTCQnkHwOhF8YtsTJh9FZmjbjZvT6q8QpToP5mlc88LYS_3pX1dNv1lVf3jCzcu2ai94sWhDxs2WxCLQcQm4rw0e04LFg34MkLTesrYxTyvR8XTZ7e5-cIYRpygs/s773/kurva+bergerak8.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"kurva sinus dan cosinus\" border=\"0\" data-original-height=\"510\" data-original-width=\"773\" height=\"211\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_131sp21eN436FAPWTCQnkHwOhF8YtsTJh9FZmjbjZvT6q8QpToP5mlc88LYS_3pX1dNv1lVf3jCzcu2ai94sWhDxs2WxCLQcQm4rw0e04LFg34MkLTesrYxTyvR8XTZ7e5-cIYRpygs/w320-h211/kurva+bergerak8.png\" title=\"kurva sinus dan cosinus\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"6-enable-animation\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-enable-animation\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Enable Animation\"\u003e\u003c/a\u003e6. Enable Animation\u003c/h3\u003e\u003cp\u003eActivate the animation by placing the cursor on the second circle *slider/glide*, right click and click *Animation On*, then both curves should move continuously according to changes in the numbers in *slider/glide*. Look at the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-VGjA40IrC5tM2XeSo4v_VdI3u0TolGlYrA4NBkFYjLeOGyGes3-JA-aGMEIqCQQ3sabBSz56oaBYhieaw71JRjoTdjEfRkcWcOrn0ppUwOJNG94BtjkCErTwSuXyfLX9VwQBniNEIbk/s773/kurva+bergerak9.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"animation on pada kurva sinus dan cosinus\" border=\"0\" data-original-height=\"514\" data-original-width=\"773\" height=\"213\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-VGjA40IrC5tM2XeSo4v_VdI3u0TolGlYrA4NBkFYjLeOGyGes3-JA-aGMEIqCQQ3sabBSz56oaBYhieaw71JRjoTdjEfRkcWcOrn0ppUwOJNG94BtjkCErTwSuXyfLX9VwQBniNEIbk/w320-h213/kurva+bergerak9.png\" title=\"animation on pada kurva\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"the-final-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-final-result\" class=\"header-mark\" aria-label=\"Direct link to section: The final result\"\u003e\u003c/a\u003eThe final result\u003c/h3\u003e\u003cp\u003eNext, tidy up the display of curves, sliders, graphs and so on. Are the results the same as what shown below? Click **play icon** at the bottom left or move it *slider a* And *slider b *to see the waves ~~Love~~ sine curve and cosine curve.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/bptxdh4w/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/false/ctl/false\" style=\"border: 0px; width: 100%; max-width: 650px;\" title=\"Kurva Bergerak\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThat's the article for creating and animating sine curves and cosine curves in GeoGebra. If you have suggestions or corrections, please leave them in the comments. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Leveling Mathematical Equations", "date_published" : "2025-12-14T06:09:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/meratakan-persamaan-matematika/", "url" : "https://www.aantriono.com/en/meratakan-persamaan-matematika/", "summary": "When we create a mathematical document, of course we want the mathematical notations in it to look neat and beautiful. One thing that sometimes causes problems is how to align the mathematical equations in the document. $\\LaTeX$ has provided various ways to flatten the mathematical equations created, including:\nHow to Flatten Mathematical Equations in LaTeX 1. Using commands eqnarrayIf using command *eqnarray*, then the equation created also displays the equation number. For example, consider the following code.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eWhen we create a mathematical document, of course we want the mathematical notations in it to look neat and beautiful. One thing that sometimes causes problems is how to align the mathematical equations in the document. $\\LaTeX$ has provided various ways to flatten the mathematical equations created, including:\u003c/p\u003e\n\u003ch2 id=\"how-to-flatten-mathematical-equations-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-flatten-mathematical-equations-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: How to Flatten Mathematical Equations in LaTeX\"\u003e\u003c/a\u003eHow to Flatten Mathematical Equations in LaTeX\u003c/h2\u003e\u003ch3 id=\"1-using-commands-eqnarray\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-using-commands-eqnarray\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Using commands eqnarray\"\u003e\u003c/a\u003e1. Using commands eqnarray\u003c/h3\u003e\u003cp\u003eIf using command *eqnarray*, then the equation created also displays the equation number. For example, consider the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eeqnarray\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 9\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e27\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e (3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2)\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 3\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e4x-6\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 4x-6\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 4x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e9\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e\u003cspan class=\"k\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e9\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eeqnarray\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is shown in the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdhJ1xKnu-52CbhlQm8yeDCYbBoofn1SXTLeMH8eYKp-bMu-6WLxu5stCsorkJcSrHM53poLQIntKUKClcM2S5eD6Flu2mFucFfYDCA53juSuJTYgXGHNnrG3MN_8dcU5F8lpdaf2D_1U/s1178/rata1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"mode eqnarray di LaTeX\" border=\"0\" data-original-height=\"417\" data-original-width=\"1178\" height=\"113\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdhJ1xKnu-52CbhlQm8yeDCYbBoofn1SXTLeMH8eYKp-bMu-6WLxu5stCsorkJcSrHM53poLQIntKUKClcM2S5eD6Flu2mFucFfYDCA53juSuJTYgXGHNnrG3MN_8dcU5F8lpdaf2D_1U/w320-h113/rata1.png\" title=\"mode eqnarray di LaTeX\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"2-using-commands-eqnarray\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-using-commands-eqnarray\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Using commands eqnarray*\"\u003e\u003c/a\u003e2. Using commands eqnarray*\u003c/h3\u003e\u003cp\u003eAs seen in the document results using the command *eqnarray*, there is numbering in the formula. To remove the numbering, we can use the command *eqnarray**. For example, consider the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eeqnarray*\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 9\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e27\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e (3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2)\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 3\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e4x-6\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 4x-6\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 4x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e9\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e\u003cspan class=\"k\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e9\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eeqnarray*\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is shown in the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBnJw8niA1wDs4NwTfukKu6HGouTuXt_qhOXbFIors6WNMFs5_o1nWRZrnOwyZcBhqD2wt8El4XwtWKxk01eJzNqsUBLXyrbhyphenhyphen6BxYnCe4Hu3zLENPytuYYYDWjcF64XDJbf1Uuh0scJU/s856/rata2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"mode eqnarray* di LaTeX\" border=\"0\" data-original-height=\"413\" data-original-width=\"856\" height=\"154\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBnJw8niA1wDs4NwTfukKu6HGouTuXt_qhOXbFIors6WNMFs5_o1nWRZrnOwyZcBhqD2wt8El4XwtWKxk01eJzNqsUBLXyrbhyphenhyphen6BxYnCe4Hu3zLENPytuYYYDWjcF64XDJbf1Uuh0scJU/w320-h154/rata2.png\" title=\"mode eqnarray* di LaTeX\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"3-setting-the-mark-position--on-command-eqnarray\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-setting-the-mark-position--on-command-eqnarray\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Setting the Mark Position \u0026ldquo;\u0026amp;\u0026rdquo; on command eqnarray*\"\u003e\u003c/a\u003e3. Setting the Mark Position \u0026ldquo;\u0026amp;\u0026rdquo; on command eqnarray*\u003c/h3\u003e\u003cp\u003eIn the example using the command *eqnarray* And *eqnarray** above, it appears that the arrows are still not aligned from top to bottom. To align it we will change the position of the \"\u0026\", and keep using the command *eqnarray**. Look at the following code example.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eeqnarray*\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 9\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=27\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e (3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2)\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 3\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e4x-6\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 4x-6\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 4x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=9\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\dfrac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e9\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eeqnarray*\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is shown in the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE-Ap0SIfF9ICEd4dm-250S9R5mu4R27oSqzq0E7JX-uMd3sIXK1_X36iW571Bu0w8G219YKQhO7b4xbmQzOe4pD61mo93doznU3WGzRaYkZU-jYoLDH0GLe-_exedqugX_dDnso2DcY0/s979/rata3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"mode eqnarray* di LaTeX\" border=\"0\" data-original-height=\"416\" data-original-width=\"979\" height=\"136\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE-Ap0SIfF9ICEd4dm-250S9R5mu4R27oSqzq0E7JX-uMd3sIXK1_X36iW571Bu0w8G219YKQhO7b4xbmQzOe4pD61mo93doznU3WGzRaYkZU-jYoLDH0GLe-_exedqugX_dDnso2DcY0/w320-h136/rata3.png\" title=\"mode eqnarray* di LaTeX\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"4-using-commands-align\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-using-commands-align\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Using commands align\"\u003e\u003c/a\u003e4. Using commands align\u003c/h3\u003e\u003cp\u003eApart from using commands *eqnarray* And *eqnarray**, we can also use commands *align*. Look at the following code example.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ealign*\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 9\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=27\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e (3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2)\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e2x-3\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 3\u003cspan class=\"nb\"\u003e^{\u003c/span\u003e4x-6\u003cspan class=\"nb\"\u003e}\u0026amp;\u003c/span\u003e=3\u003cspan class=\"nb\"\u003e^\u003c/span\u003e3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 4x-6\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=3\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e 4x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=9\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\Leftrightarrow\u003c/span\u003e x\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"k\"\u003e\\dfrac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e9\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ealign*\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document is shown in the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjB8J0p9YWpGHHsb18av6W56kqg9q8vab5b-eTwtvAwdRAphxurDEv9R8J6hh5eHRsRBbxX_jyDWjDCg0KgCtNt0K1eIWl-skm_hyphenhyphen84oUA99_moyur3Vq5R7R36QctiGDAnF0M7zeEfjo/s971/rata4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"mode align di LaTeX\" border=\"0\" data-original-height=\"423\" data-original-width=\"971\" height=\"139\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjB8J0p9YWpGHHsb18av6W56kqg9q8vab5b-eTwtvAwdRAphxurDEv9R8J6hh5eHRsRBbxX_jyDWjDCg0KgCtNt0K1eIWl-skm_hyphenhyphen84oUA99_moyur3Vq5R7R36QctiGDAnF0M7zeEfjo/w320-h139/rata4.png\" title=\"mode align di LaTeX\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003ch3 id=\"5-using-commands-tabular\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-using-commands-tabular\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Using commands tabular\"\u003e\u003c/a\u003e5. Using commands tabular\u003c/h3\u003e\u003cp\u003eThen to flatten the formula, we can also use tabular commands. Pay attention to the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003elll\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e(1/y)dy \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\lambda\u003c/span\u003e\u003cspan class=\"nb\"\u003e dt\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e ln y \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\lambda\u003c/span\u003e\u003cspan class=\"nb\"\u003e t \u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003ec\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\Leftrightarrow\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e y \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e=\u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ec.e^{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\lambda\u003c/span\u003e\u003cspan class=\"nb\"\u003e t}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe resulting document can be seen in the following image.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7fyAnRrTCa1mF9tIjdIgmsuHt76I4p6f4KDAalU62Lq9a9cgWDiFpdrmME3qJGRtwrws3R85miugbrSDSXztyfrwPKbmTUBDAioaxfQVW-95Wcsfn2tzIkWrdyns29eHINMAVJ4Fhxhg/s923/rata5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\n \u003cimg alt=\"mode tabular di LaTeX\" border=\"0\" data-original-height=\"354\" data-original-width=\"923\" height=\"123\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7fyAnRrTCa1mF9tIjdIgmsuHt76I4p6f4KDAalU62Lq9a9cgWDiFpdrmME3qJGRtwrws3R85miugbrSDSXztyfrwPKbmTUBDAioaxfQVW-95Wcsfn2tzIkWrdyns29eHINMAVJ4Fhxhg/w320-h123/rata5.png\" title=\"mode tabular di LaTeX\" width=\"320\" /\u003e\n\u003c/a\u003e\n\u003cp\u003eThat's the article on how to flatten mathematical equations in $\\LaTeX$. If you have suggestions or corrections, please leave them in the comments. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Writing Mathematical Notation in LaTeX", "date_published" : "2025-12-14T05:56:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menulis-notasi-matematika-di-latex/", "url" : "https://www.aantriono.com/en/menulis-notasi-matematika-di-latex/", "summary": "As is well known, TeX\\TeXTE​X was designed to produce beautiful books, especially books containing a lot of mathematics. Because of that, LaTeX\\LaTeXLATE​X is very strong for preparing documents with mathematical notation. To write mathematical symbols and formulas, we need to use the mathematical environment.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eAs is well known, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e was designed to produce beautiful books, especially books containing a lot of mathematics. Because of that, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is very strong for preparing documents with mathematical notation. To write mathematical symbols and formulas, we need to use the \u003cem\u003emathematical environment\u003c/em\u003e.\u003c/p\u003e\n\u003ch2 id=\"displaying-mathematical-formulas\" class=\"headerLink\"\u003e\n \u003ca href=\"#displaying-mathematical-formulas\" class=\"header-mark\" aria-label=\"Direct link to section: Displaying Mathematical Formulas\"\u003e\u003c/a\u003eDisplaying Mathematical Formulas\u003c/h2\u003e\u003cp\u003eMathematical formulas in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e can be written between \u003ccode\u003e$...$\u003c/code\u003e or \u003ccode\u003e$$...$$\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eThe difference is:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eFormulas between \u003ccode\u003e$...$\u003c/code\u003e or \u003ccode\u003e\\(...\\)\u003c/code\u003e appear as part of a sentence. This is called an \u003cem\u003einline equation\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eFormulas between \u003ccode\u003e$$...$$\u003c/code\u003e or \u003ccode\u003e\\[...\\]\u003c/code\u003e appear as stand-alone formulas. This is called a \u003cem\u003edisplayed equation\u003c/em\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eIf you want numbered equations, use the \u003ccode\u003eequation\u003c/code\u003e environment.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eInline formula: \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ea^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003eb^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"nb\"\u003ec^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"nb\"\u003e^\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"nb\"\u003e x^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\,\u003c/span\u003e\u003cspan class=\"nb\"\u003e dx\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eequation\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eE = mc\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eequation\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output can be seen in the following image.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif8ZVVK61rBMtazh84KMRgfjkaw8bfO_EG-Cq8dQ_UzWlvQ9oH2wF-kBj2NLVqEBEzD_sSY-hQlDZy0CTcbaHZycYma2jwnPeiizCHxkvSYl4qd-e3fnYzyzw3pq8gjaNhl-lS20NHUBM/s1228/rumus1.png' alt=\"Inline and displayed formulas in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"subscript-and-superscript\" class=\"headerLink\"\u003e\n \u003ca href=\"#subscript-and-superscript\" class=\"header-mark\" aria-label=\"Direct link to section: Subscript and Superscript\"\u003e\u003c/a\u003eSubscript and Superscript\u003c/h2\u003e\u003cp\u003eA \u003cem\u003esubscript\u003c/em\u003e is written below a symbol and usually uses \u003ccode\u003e_\u003c/code\u003e.\u003cbr\u003e\nA \u003cem\u003esuperscript\u003c/em\u003e is written above a symbol and usually uses \u003ccode\u003e^\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ex_\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\ \u003c/span\u003e\u003cspan class=\"nb\"\u003ea_n,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\ \u003c/span\u003e\u003cspan class=\"nb\"\u003ey^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\ \u003c/span\u003e\u003cspan class=\"nb\"\u003ex^{n\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOwfTd0tH9_CkaJRqpiyojDL1x8fOhwYgWdEmrvra8VSNFFYpETj0pJLE7iYdJika-tJCeNwYwwylMtEAnZ_2OLduz6nfxGETHwI83U92h22X2bUP84n4bN5E7JNNQ2vzX4zInryD6TKQ/s1200/rumus2.png' alt=\"Subscript and superscript in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"binomial-and-division-style-notation\" class=\"headerLink\"\u003e\n \u003ca href=\"#binomial-and-division-style-notation\" class=\"header-mark\" aria-label=\"Direct link to section: Binomial and Division-Style Notation\"\u003e\u003c/a\u003eBinomial and Division-Style Notation\u003c/h2\u003e\u003cp\u003eTo write binomial notation, use \u003ccode\u003e\\binom{...}{...}\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\binom\u003c/span\u003e\u003cspan class=\"nb\"\u003e{n}{r}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgONO8kkg-i9cmllj_-EF9yg58-mdcAB5ZwyaFVfWc7mQ8jVFNWL8nJAdtzsCjfd2NQ1EmSTm9CnN1mJAyHm-0RkesBPJqpOHUQ28gs-sivuJJG6Bgfx0tX2QwtzC4ynJheqsTs_zvn0ZA/s1081/rumus3.png' alt=\"Binomial notation in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"integrals-roots-and-fractions\" class=\"headerLink\"\u003e\n \u003ca href=\"#integrals-roots-and-fractions\" class=\"header-mark\" aria-label=\"Direct link to section: Integrals, Roots, and Fractions\"\u003e\u003c/a\u003eIntegrals, Roots, and Fractions\u003c/h2\u003e\u003cp\u003eTo write:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eintegrals, use \u003ccode\u003e\\int\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eroots, use \u003ccode\u003e\\sqrt{...}\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003efractions, use \u003ccode\u003e\\frac{...}{...}\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e x^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\,\u003c/span\u003e\u003cspan class=\"nb\"\u003e dx,\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\qquad\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\sqrt\u003c/span\u003e\u003cspan class=\"nb\"\u003e{x\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"nb\"\u003e},\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\qquad\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{a\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003eb}{c\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003ed}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPd2IGZ9wjr8t3k10Aqx7D_n3pRD7GYYxHrEIqc2FPIwIBKQIIYCXtiaOTc1BkceF_wW95rAFJNxXjycXV-JaqRvAKz8tbXXPAHFPm0ZlzAlCHVwJL1-5nQ3Pld9HqZcwK1OSdDmFSQtQ/s876/rumus4.png' alt=\"Integral, root, and fraction notation in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"matrix\" class=\"headerLink\"\u003e\n \u003ca href=\"#matrix\" class=\"header-mark\" aria-label=\"Direct link to section: Matrix\"\u003e\u003c/a\u003eMatrix\u003c/h2\u003e\u003cp\u003eTo write matrices, we can use parentheses or square brackets.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{pmatrix}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u0026amp; \u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"m\"\u003e3\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u0026amp; \u003c/span\u003e\u003cspan class=\"m\"\u003e4\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{pmatrix}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\qquad\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{bmatrix}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003ea \u0026amp; b \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003ec \u0026amp; d\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{bmatrix}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOHeDIwV1-Txg8nVjt_6yox1VTqo57YHHTDwx6PB1nSb5RtBBoMdvxeOSGk6QJV460MfPVW2yVTUXF3pABXtRUzapm6lT-DwUIc42SGwHqEpv5KyEeVAwNgE2s-2x2Hnac540qPriqjkI/s1021/matriks.png' alt=\"Matrix in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"limit\" class=\"headerLink\"\u003e\n \u003ca href=\"#limit\" class=\"header-mark\" aria-label=\"Direct link to section: Limit\"\u003e\u003c/a\u003eLimit\u003c/h2\u003e\u003cp\u003eTo write limits, use \u003ccode\u003e\\lim\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\lim\u003c/span\u003e\u003cspan class=\"nb\"\u003e_{x \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\to\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\sin\u003c/span\u003e\u003cspan class=\"nb\"\u003e x}{x}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLIPedz0lBI1oeVS7V-tFMXXOd9WqJkz01A_OzNTq4FkhtZYu45wyFX-YWqeGkY7L-2jO6NZStBVBd5q4s2ZCQZCWIq7Q6XtY-VlhJLITDWVw8aoPa_tIHc2BpFG-zhvW0qWBpI4BbqLQ/s1031/limit.png' alt=\"Limit in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"sigma\" class=\"headerLink\"\u003e\n \u003ca href=\"#sigma\" class=\"header-mark\" aria-label=\"Direct link to section: Sigma\"\u003e\u003c/a\u003eSigma\u003c/h2\u003e\u003cp\u003eTo write sigma notation, use \u003ccode\u003e\\sum\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\sum\u003c/span\u003e\u003cspan class=\"nb\"\u003e_{i\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"nb\"\u003e}^{n} i^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCZRS3MqCWjboULJm_rOpxOsRPbavgA4Ck5q3dNTKoH-2hTXYl5cnI0FoxapWqPRbXrZDresV58VbBwiWH3LT6Ye8R9sDf9398lqTal5ZH1S7Vb3YFAWg80Bbdx4RAtwvgqnDvvIR0xuI/s1011/sigma.png' alt=\"Sigma in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"systems-of-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#systems-of-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Systems of Equations\"\u003e\u003c/a\u003eSystems of Equations\u003c/h2\u003e\u003cp\u003eSystems of equations are commonly written with curly brackets. For this, we can use the \u003ccode\u003eamsmath\u003c/code\u003e or \u003ccode\u003esysteme\u003c/code\u003e package.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{cases}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003ex\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003ey\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"m\"\u003e5\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\\\\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003ex\u003c/span\u003e\u003cspan class=\"o\"\u003e-\u003c/span\u003e\u003cspan class=\"nb\"\u003ey\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"m\"\u003e1\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{cases}\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1kRh-rF_scRSEvl0mToCtJr-vBH8zFU7E5zmnkSzR7IPNK7xq1djyD6hK8JW-xkItkdIQEmDk1_kftThZyfaS4hR5C1XJmp-g4cF5PxUIo5hUsmDCyYuGY2HekwTbYHBvQG7KqDU0Ilg/s1019/sistem+persamaan.png' alt=\"System of equations in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"brackets\" class=\"headerLink\"\u003e\n \u003ca href=\"#brackets\" class=\"header-mark\" aria-label=\"Direct link to section: Brackets\"\u003e\u003c/a\u003eBrackets\u003c/h2\u003e\u003cp\u003eIn mathematical formulas, brackets are often used to separate expressions.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\left\u003c/span\u003e\u003cspan class=\"o\"\u003e(\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{a\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003eb}{c\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003ed} \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\right\u003c/span\u003e\u003cspan class=\"o\"\u003e)\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\quad\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\left\u003c/span\u003e\u003cspan class=\"o\"\u003e[\u003c/span\u003e\u003cspan class=\"nb\"\u003e x^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003ey^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\right\u003c/span\u003e\u003cspan class=\"o\"\u003e]\u003c/span\u003e\u003cspan class=\"nb\"\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\quad\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\left\\{\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{x}{y} \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\right\\}\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM5JCrSod2LsxcOHi88mdw9FTrSO0Qj-5RkXHuUAIrce6mZvE47JV1m7cecRLaq0sXf2vIVkycAkhC1A4sVCt-ZvLSnFq92La78ddajD93U01nl06gQzs5FKzgv86-GWguN8SaFRnRynM/s931/tanda+kurung.png' alt=\"Brackets in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eBesides the examples above, there are many more techniques for writing mathematics in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. If you want to study the symbols in more detail, the documentation can be downloaded \u003ca href=\"https://mirror.las.iastate.edu/tex-archive/info/symbols/comprehensive/symbols-letter.pdf\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehere\u003c/a\u003e.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat is a short guide to writing mathematical notation in LaTeX. If you have suggestions or corrections, please leave them in the comments.\u003c/p\u003e\n", "language": "en" }, { "title" : "Writing in LaTeX", "date_published" : "2025-12-14T05:38:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/tulisan-dalam-latex/", "url" : "https://www.aantriono.com/en/tulisan-dalam-latex/", "summary": " Typography and Text Styling in LaTeXLaTeX\\LaTeXLATE​X gives you a lot of control over how text appears in a document: font family, weight, shape, size, color, boxed text, and even rotated text. These features are useful when you want cleaner and more expressive document formatting.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"typography-and-text-styling-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#typography-and-text-styling-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Typography and Text Styling in LaTeX\"\u003e\u003c/a\u003eTypography and Text Styling in LaTeX\u003c/h2\u003e\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e gives you a lot of control over how text appears in a document: font family, weight, shape, size, color, boxed text, and even rotated text. These features are useful when you want cleaner and more expressive document formatting.\u003c/p\u003e\n\u003ch3 id=\"text-attributes-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#text-attributes-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Text Attributes in LaTeX\"\u003e\u003c/a\u003eText Attributes in LaTeX\u003c/h3\u003e\u003cp\u003eText in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is usually described through five main attributes:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eEncoding\u003c/strong\u003e: how characters are represented.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eFamily\u003c/strong\u003e: roman, sans serif, or typewriter.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSeries\u003c/strong\u003e: weight such as normal or bold.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eShape\u003c/strong\u003e: upright, italic, slanted, or small caps.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSize\u003c/strong\u003e: from \u003ccode\u003e\\tiny\u003c/code\u003e up to \u003ccode\u003e\\Huge\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"commands-and-declarations\" class=\"headerLink\"\u003e\n \u003ca href=\"#commands-and-declarations\" class=\"header-mark\" aria-label=\"Direct link to section: Commands and Declarations\"\u003e\u003c/a\u003eCommands and Declarations\u003c/h3\u003e\u003cp\u003eMost text styles in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e can be written in two forms:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ecommand form, for example \u003ccode\u003e\\textbf{text}\u003c/code\u003e,\u003c/li\u003e\n\u003cli\u003edeclaration form, for example \u003ccode\u003e{\\bfseries text}\u003c/code\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eCommon examples:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eStyle\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eCommand\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eDeclaration\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eroman\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textrm{roman}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\rmfamily roman}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003esans serif\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textsf{sans serif}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\sffamily sans serif}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003etypewriter\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\texttt{typewriter}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\ttfamily typewriter}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003emedium\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textmd{medium}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\mdseries medium}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eboldface\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textbf{boldface}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\bfseries boldface}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eupright\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textup{upright}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\upshape upright}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eitalic\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textit{italic}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\itshape italic}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eslanted\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textsl{slanted}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\slshape slanted}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003esmall caps\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\textsc{small caps}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e{\\scshape small caps}\u003c/code\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eExample output for font family and shape settings:\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmxLwOWwNQ1HOzz9bI8ODjUda3YM_ecRCXxtU5J9VaLDMiXcv7mQ0YVu-YWi9pq8HbW_Sm0SMrlQWnP8NBMVE6oap_WPbp8eELI2xajxZqLI0bX4fWsmi_Th3F98rG_zS8WNof9x6BW60/w320-h159/font1.png\" title=\"Example of font settings in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmxLwOWwNQ1HOzz9bI8ODjUda3YM_ecRCXxtU5J9VaLDMiXcv7mQ0YVu-YWi9pq8HbW_Sm0SMrlQWnP8NBMVE6oap_WPbp8eELI2xajxZqLI0bX4fWsmi_Th3F98rG_zS8WNof9x6BW60/w320-h159/font1.png\" data-sub-html=\"\u003ch2\u003eExample of font settings in LaTeX\u003c/h2\u003e\u003cp\u003eExample of font settings in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmxLwOWwNQ1HOzz9bI8ODjUda3YM_ecRCXxtU5J9VaLDMiXcv7mQ0YVu-YWi9pq8HbW_Sm0SMrlQWnP8NBMVE6oap_WPbp8eELI2xajxZqLI0bX4fWsmi_Th3F98rG_zS8WNof9x6BW60/w320-h159/font1.png' alt=\"Example of font settings in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eExample of font settings in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmxLwOWwNQ1HOzz9bI8ODjUda3YM_ecRCXxtU5J9VaLDMiXcv7mQ0YVu-YWi9pq8HbW_Sm0SMrlQWnP8NBMVE6oap_WPbp8eELI2xajxZqLI0bX4fWsmi_Th3F98rG_zS8WNof9x6BW60/s994/font1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmxLwOWwNQ1HOzz9bI8ODjUda3YM_ecRCXxtU5J9VaLDMiXcv7mQ0YVu-YWi9pq8HbW_Sm0SMrlQWnP8NBMVE6oap_WPbp8eELI2xajxZqLI0bX4fWsmi_Th3F98rG_zS8WNof9x6BW60/s994/font1.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"font-size\" class=\"headerLink\"\u003e\n \u003ca href=\"#font-size\" class=\"header-mark\" aria-label=\"Direct link to section: Font Size\"\u003e\u003c/a\u003eFont Size\u003c/h3\u003e\u003cp\u003eBy default, a basic \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document often uses \u003ccode\u003e10pt\u003c/code\u003e text. You can change the size with these commands:\u003c/p\u003e\n\u003cp\u003e\u003ccode\u003e\\tiny, \\scriptsize, \\footnotesize, \\small, \\normalsize, \\large, \\Large, \\LARGE, \\huge, \\Huge\u003c/code\u003e\u003c/p\u003e\n\u003cp\u003eExample of size variations:\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVVjUY21nRJIHFyi0ykvG32IlAcwyiEsUgQpscKgJ2AvLOfuPOkivm2M_YDwOqoczIbe5l03dqyHSETB7ZDeqahh8xE2W1gxzUvCPwlFmhuVPDMiK6G2VKPWqE1tYxUeMjN_xN4JMnOkA/w320-h129/font2.png\" title=\"Example of font sizes in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVVjUY21nRJIHFyi0ykvG32IlAcwyiEsUgQpscKgJ2AvLOfuPOkivm2M_YDwOqoczIbe5l03dqyHSETB7ZDeqahh8xE2W1gxzUvCPwlFmhuVPDMiK6G2VKPWqE1tYxUeMjN_xN4JMnOkA/w320-h129/font2.png\" data-sub-html=\"\u003ch2\u003eExample of font sizes in LaTeX\u003c/h2\u003e\u003cp\u003eExample of font sizes in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVVjUY21nRJIHFyi0ykvG32IlAcwyiEsUgQpscKgJ2AvLOfuPOkivm2M_YDwOqoczIbe5l03dqyHSETB7ZDeqahh8xE2W1gxzUvCPwlFmhuVPDMiK6G2VKPWqE1tYxUeMjN_xN4JMnOkA/w320-h129/font2.png' alt=\"Example of font sizes in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eExample of font sizes in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVVjUY21nRJIHFyi0ykvG32IlAcwyiEsUgQpscKgJ2AvLOfuPOkivm2M_YDwOqoczIbe5l03dqyHSETB7ZDeqahh8xE2W1gxzUvCPwlFmhuVPDMiK6G2VKPWqE1tYxUeMjN_xN4JMnOkA/s1126/font2.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVVjUY21nRJIHFyi0ykvG32IlAcwyiEsUgQpscKgJ2AvLOfuPOkivm2M_YDwOqoczIbe5l03dqyHSETB7ZDeqahh8xE2W1gxzUvCPwlFmhuVPDMiK6G2VKPWqE1tYxUeMjN_xN4JMnOkA/s1126/font2.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"coloring-text\" class=\"headerLink\"\u003e\n \u003ca href=\"#coloring-text\" class=\"header-mark\" aria-label=\"Direct link to section: Coloring Text\"\u003e\u003c/a\u003eColoring Text\u003c/h3\u003e\u003cp\u003eTo color text, you can use either of these forms:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}\u003c/span\u003e Colored text\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eor\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eColored text\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis usually requires the \u003ccode\u003excolor\u003c/code\u003e package in the preamble.\u003c/p\u003e\n\u003ch3 id=\"colored-background-for-text\" class=\"headerLink\"\u003e\n \u003ca href=\"#colored-background-for-text\" class=\"header-mark\" aria-label=\"Direct link to section: Colored Background for Text\"\u003e\u003c/a\u003eColored Background for Text\u003c/h3\u003e\u003cp\u003eTo place text on a colored background, use \u003ccode\u003e\\colorbox\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eyellow\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eText with colored background\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you also want to change the text color:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\colorbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eyellow\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ered\u003cspan class=\"nb\"\u003e}\u003c/span\u003eColored text on a background\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"text-inside-a-box\" class=\"headerLink\"\u003e\n \u003ca href=\"#text-inside-a-box\" class=\"header-mark\" aria-label=\"Direct link to section: Text Inside a Box\"\u003e\u003c/a\u003eText Inside a Box\u003c/h3\u003e\u003cp\u003eTo put text inside a simple box:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eText inside a box\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTo create a colored box:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fcolorbox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblack\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eyellow\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eText inside a colored box\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThis also relies on the \u003ccode\u003excolor\u003c/code\u003e package.\u003c/p\u003e\n\u003ch3 id=\"rotating-text\" class=\"headerLink\"\u003e\n \u003ca href=\"#rotating-text\" class=\"header-mark\" aria-label=\"Direct link to section: Rotating Text\"\u003e\u003c/a\u003eRotating Text\u003c/h3\u003e\u003cp\u003eTo rotate text, use \u003ccode\u003e\\rotatebox\u003c/code\u003e from the \u003ccode\u003egraphicx\u003c/code\u003e package.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rotatebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e45\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eRotated text\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eExample output for colored text, boxed text, and rotated text:\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4oKHBloZSD_wKQ0NBvzV040o2kpUHnPNQV6NNcJSGYOA7OJgS2DBNcosF_ZGkkjZpb83EmL2xIgQrv2cvwCP0fgT6RRxCpq-IW3d5_NISBv_RRkXQUqKcrp-ONBJxWX7vPePorXiM1Ug/w320-h132/font3.png\" title=\"Example of color and rotated text in LaTeX\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4oKHBloZSD_wKQ0NBvzV040o2kpUHnPNQV6NNcJSGYOA7OJgS2DBNcosF_ZGkkjZpb83EmL2xIgQrv2cvwCP0fgT6RRxCpq-IW3d5_NISBv_RRkXQUqKcrp-ONBJxWX7vPePorXiM1Ug/w320-h132/font3.png\" data-sub-html=\"\u003ch2\u003eExample of color and rotated text in LaTeX\u003c/h2\u003e\u003cp\u003eExample of color and rotated text in LaTeX\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4oKHBloZSD_wKQ0NBvzV040o2kpUHnPNQV6NNcJSGYOA7OJgS2DBNcosF_ZGkkjZpb83EmL2xIgQrv2cvwCP0fgT6RRxCpq-IW3d5_NISBv_RRkXQUqKcrp-ONBJxWX7vPePorXiM1Ug/w320-h132/font3.png' alt=\"Example of color and rotated text in LaTeX\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eExample of color and rotated text in LaTeX\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4oKHBloZSD_wKQ0NBvzV040o2kpUHnPNQV6NNcJSGYOA7OJgS2DBNcosF_ZGkkjZpb83EmL2xIgQrv2cvwCP0fgT6RRxCpq-IW3d5_NISBv_RRkXQUqKcrp-ONBJxWX7vPePorXiM1Ug/s1214/font3.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4oKHBloZSD_wKQ0NBvzV040o2kpUHnPNQV6NNcJSGYOA7OJgS2DBNcosF_ZGkkjZpb83EmL2xIgQrv2cvwCP0fgT6RRxCpq-IW3d5_NISBv_RRkXQUqKcrp-ONBJxWX7vPePorXiM1Ug/s1214/font3.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eTypography control in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is rich enough for both standard documents and more styled layouts. Once you understand font family, size, color, boxed text, and rotation, it becomes much easier to produce documents that look clear and polished.\u003c/p\u003e\n", "language": "en" }, { "title" : "Environment", "date_published" : "2025-12-14T05:33:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/environment/", "url" : "https://www.aantriono.com/en/environment/", "summary": " Environment (Command Scope) in LaTeXTo format special parts of a document, LaTeX provides many environments. An environment controls a specific layout or behavior, such as lists, paragraph alignment, quotations, verses, or displayed mathematics.\nIn general, an environment is written in this form:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"environment-command-scope-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#environment-command-scope-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Environment (Command Scope) in LaTeX\"\u003e\u003c/a\u003eEnvironment (Command Scope) in LaTeX\u003c/h2\u003e\u003cp\u003eTo format special parts of a document, LaTeX provides many \u003cem\u003eenvironments\u003c/em\u003e. An environment controls a specific layout or behavior, such as lists, paragraph alignment, quotations, verses, or displayed mathematics.\u003c/p\u003e\n\u003cp\u003eIn general, an environment is written in this form:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenvironment-name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenvironment-name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eSome of the most common environments are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003eitemize\u003c/code\u003e for unordered lists\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eenumerate\u003c/code\u003e for numbered lists\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003edescription\u003c/code\u003e for lists with keyword labels\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003equote\u003c/code\u003e for short quotations\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003equotation\u003c/code\u003e for longer quotations with multiple paragraphs\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003everse\u003c/code\u003e for song lyrics or poetry\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003edisplaymath\u003c/code\u003e for displayed mathematical expressions\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"item-lists\" class=\"headerLink\"\u003e\n \u003ca href=\"#item-lists\" class=\"header-mark\" aria-label=\"Direct link to section: Item Lists\"\u003e\u003c/a\u003eItem Lists\u003c/h3\u003e\u003cp\u003eYou can create lists with \u003ccode\u003eitemize\u003c/code\u003e, \u003ccode\u003eenumerate\u003c/code\u003e, and \u003ccode\u003edescription\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eitemize\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Apple\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Orange\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Mango\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eitemize\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Wake up\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Have breakfast\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e Go to school\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eenumerate\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edescription\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e\u003cspan class=\"na\"\u003e[Triangle]\u003c/span\u003e A polygon with three sides.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\item\u003c/span\u003e\u003cspan class=\"na\"\u003e[Square]\u003c/span\u003e A shape with four equal sides.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edescription\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRMchFzEoGprC4NsuXCafJ37ksJqxY3C2UToeRx6FtWyFwUIZ8prEKkTKfrvAflFS5FeUIzvKcZI_Hsg-Ab-UMa0OsQd0KrPpJ0PAoaMxkZB_U3B_4uqD40m1D32sA4xV5J15LsDKS7Aw/s1028/environment1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"enumerate, itemize, description in LaTeX\" border=\"0\" data-original-height=\"492\" data-original-width=\"1028\" height=\"153\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRMchFzEoGprC4NsuXCafJ37ksJqxY3C2UToeRx6FtWyFwUIZ8prEKkTKfrvAflFS5FeUIzvKcZI_Hsg-Ab-UMa0OsQd0KrPpJ0PAoaMxkZB_U3B_4uqD40m1D32sA4xV5J15LsDKS7Aw/w320-h153/environment1.png\" title=\"enumerate, itemize, description\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"flushleft-flushright-center\" class=\"headerLink\"\u003e\n \u003ca href=\"#flushleft-flushright-center\" class=\"header-mark\" aria-label=\"Direct link to section: Flushleft, Flushright, Center\"\u003e\u003c/a\u003eFlushleft, Flushright, Center\u003c/h3\u003e\u003cp\u003eUse \u003ccode\u003eflushleft\u003c/code\u003e, \u003ccode\u003eflushright\u003c/code\u003e, and \u003ccode\u003ecenter\u003c/code\u003e to align paragraphs.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushleft\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis paragraph is left aligned.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushleft\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis paragraph is centered.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecenter\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushright\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis paragraph is right aligned.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushright\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_ux8MjGwPmhe7wjhtKhgmkBPUmxc7I5y3oFH3OmyB9WUb1rSaII_jbXPvchpe2xHo_T_COHYD9QJR_sSbH0284_G0mIia5NNbQyXCjTu7sNu3DylkN2ZIMVEzrsXnIZuaWUYiUKpz7U0/s1225/environment2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"flushleft, center, flushright in LaTeX\" border=\"0\" data-original-height=\"495\" data-original-width=\"1225\" height=\"129\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_ux8MjGwPmhe7wjhtKhgmkBPUmxc7I5y3oFH3OmyB9WUb1rSaII_jbXPvchpe2xHo_T_COHYD9QJR_sSbH0284_G0mIia5NNbQyXCjTu7sNu3DylkN2ZIMVEzrsXnIZuaWUYiUKpz7U0/w320-h129/environment2.png\" title=\"flushleft, center, flushright in LaTeX\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"quotations-and-verses\" class=\"headerLink\"\u003e\n \u003ca href=\"#quotations-and-verses\" class=\"header-mark\" aria-label=\"Direct link to section: Quotations and Verses\"\u003e\u003c/a\u003eQuotations and Verses\u003c/h3\u003e\u003cp\u003eUse \u003ccode\u003equote\u003c/code\u003e for short quotations and \u003ccode\u003equotation\u003c/code\u003e for longer ones.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003equote\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eLearning a little every day is better than never starting.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003equote\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003equotation\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eMathematics is not only about calculation, but also about thinking.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThat is why steady practice matters.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003equotation\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDhb9Yz3p88k2uC6TBUh6kWzSsY2dies6h00P7qnLpWMKa-WdXRkkY56_AkskXu6zCNe3oWxpLC-B1HHODqsbnJ9GXhDgNMegcX1Fl53JzEB9AgbyeDvBZg54RciB1KFZb0QA5P1QNSMI/s1229/environment5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"quotation in LaTeX\" border=\"0\" data-original-height=\"476\" data-original-width=\"1229\" height=\"124\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDhb9Yz3p88k2uC6TBUh6kWzSsY2dies6h00P7qnLpWMKa-WdXRkkY56_AkskXu6zCNe3oWxpLC-B1HHODqsbnJ9GXhDgNMegcX1Fl53JzEB9AgbyeDvBZg54RciB1KFZb0QA5P1QNSMI/w320-h124/environment5.png\" title=\"quotation in LaTeX\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eFor poetry or lyrics, use \u003ccode\u003everse\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003everse\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eKnowledge grows from day to day,\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSmall steps still can lead the way,\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eKeep on learning, keep on trying,\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eGood results will soon be showing.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003everse\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvZYnfFg2IexHctcrQz-X67L795HOdtsNPuqPYOvT9zfYYmUeAVtGQzti2KzLpTKYjD7HcYB1j7M-tLn-vzzYDm0ZI_TqXcEogLr6ozcA3S_PHGRWRJWnUDlNNx5WP2WL2kW7thfiIISQ/s1219/environment4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"verse environment in LaTeX\" border=\"0\" data-original-height=\"471\" data-original-width=\"1219\" height=\"124\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvZYnfFg2IexHctcrQz-X67L795HOdtsNPuqPYOvT9zfYYmUeAVtGQzti2KzLpTKYjD7HcYB1j7M-tLn-vzzYDm0ZI_TqXcEogLr6ozcA3S_PHGRWRJWnUDlNNx5WP2WL2kW7thfiIISQ/w320-h124/environment4.png\" title=\"verse environment in LaTeX\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eDisplayed math environments deserve a separate article because mathematical notation usually needs a longer and more focused explanation.\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat is a short introduction to environments in LaTeX. Once you understand them, it becomes much easier to structure lists, align text, and format quotations cleanly.\u003c/p\u003e\n", "language": "en" }, { "title" : "Creating Tables in LaTeX", "date_published" : "2023-03-01T15:25:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/membuat-tabel-di-latex/", "url" : "https://www.aantriono.com/en/membuat-tabel-di-latex/", "summary": "Creating tables in LaTeX\\LaTeXLATE​X is different from creating them in Excel or a spreadsheet application. In spreadsheets, row and column borders are already visible. In LaTeX\\LaTeXLATE​X, tables are written with specific syntax so the result matches the layout you need.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eCreating tables in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is different from creating them in Excel or a spreadsheet application. In spreadsheets, row and column borders are already visible. In \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, tables are written with specific syntax so the result matches the layout you need.\u003c/p\u003e\n\u003cp\u003eThe most common environment is \u003ccode\u003etabular\u003c/code\u003e. After \u003ccode\u003e\\begin{tabular}\u003c/code\u003e, you write the column specification using:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003el\u003c/code\u003e for left alignment,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003ec\u003c/code\u003e for centered alignment,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003er\u003c/code\u003e for right alignment,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003ep{...}\u003c/code\u003e for a paragraph column with a fixed width.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eColumns are separated by \u003ccode\u003e\u0026amp;\u003c/code\u003e, and each row ends with \u003ccode\u003e\\\\\u003c/code\u003e. Horizontal lines are created with \u003ccode\u003e\\hline\u003c/code\u003e, while vertical lines are added with \u003ccode\u003e|\u003c/code\u003e inside the column specification.\u003c/p\u003e\n\u003ch3 id=\"table-without-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#table-without-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Table Without Lines\"\u003e\u003c/a\u003eTable Without Lines\u003c/h3\u003e\u003cp\u003eHere is a simple example of a table without borders:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h!]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ecllll\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Class \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e City \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Score \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII A \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Lampung \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 80 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Budi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII A \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Metro \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 78 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e3 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Citra \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII B \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Pringsewu \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 85 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e4 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Dini \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII B \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Bandar Jaya \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 88 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e5 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Eko \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII C \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Gunung Sugih \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 82 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEix7-mDjCyJZ7LE29vZ7C0oWjMOe4sSjuxwm9FUJU0pJ37QyODhuHmk8RqJfJnWYiDmJkTGTVVEsl8TG-KaTQA5_-RiaMB4VgcStnkzwQBUPS4xgf265-A-oYM4zxXlfdMbNmIAI6Q7qzY/s1053/tabel1.png' alt=\"Table without lines in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eIn \u003ccode\u003e{cllll}\u003c/code\u003e, the first column is centered and the remaining columns are left aligned. If you want visible borders between rows and columns, use vertical bars in the specification and \u003ccode\u003e\\hline\u003c/code\u003e inside the table.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h!]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|l|l|l|l|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Class \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e City \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Score \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII A \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Lampung \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 80 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Budi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII A \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Metro \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 78 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e3 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Citra \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII B \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Pringsewu \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 85 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e4 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Dini \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII B \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Bandar Jaya \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 88 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e5 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Eko \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII C \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Gunung Sugih \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 82 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWxovD6G-Qr9FYP3TRtS8eb6UGa1pdRxnDoUZ22H_oUyJlgIh0g6BxjUdDtlxEQKQxhyZsljQHmME_v4ixRYYVNMgekXwWT0GVrNZoDyAQ5iQsTZAgn_zybfnUK4LYvGdGfF1-7gsf29s/s1038/tabel2.png' alt=\"Table with lines in LaTeX\" \u003e\u003c/p\u003e\n\u003ch3 id=\"merging-table-rows\" class=\"headerLink\"\u003e\n \u003ca href=\"#merging-table-rows\" class=\"header-mark\" aria-label=\"Direct link to section: Merging Table Rows\"\u003e\u003c/a\u003eMerging Table Rows\u003c/h3\u003e\u003cp\u003eTo merge rows, use the \u003ccode\u003emultirow\u003c/code\u003e package.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emultirow\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|c|c|c|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e District \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Village \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Note \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\multirow\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e*\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"k\"\u003e\\multirow\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e*\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eBangunrejo\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Sidodadi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e A \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\cline\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3-4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Srikaton \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e B \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\cline\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3-4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Mulyojati \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e C \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe command \u003ccode\u003e\\multirow{3}{*}{Bangunrejo}\u003c/code\u003e makes the cell span three rows. To draw a partial horizontal line, use \u003ccode\u003e\\cline{3-4}\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmh8Aph-kGfWvnSm6hlrgs8MGbYv5w7_A5u8QQbfm6TDXRbAxvCNWuR7uZJ5GOrjZERHXe6AP4xEWbKmsrwwVXMyvN9z726S57K1DqaE-jVppRlu83QUHHeyEaYzcDkbMmbU8tQMK1v7k/s1141/tabel3.png' alt=\"Merging rows in LaTeX tables\" \u003e\u003c/p\u003e\n\u003ch3 id=\"merging-table-columns\" class=\"headerLink\"\u003e\n \u003ca href=\"#merging-table-columns\" class=\"header-mark\" aria-label=\"Direct link to section: Merging Table Columns\"\u003e\u003c/a\u003eMerging Table Columns\u003c/h3\u003e\u003cp\u003eTo merge columns, use \u003ccode\u003e\\multicolumn\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|c|c|c|c|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\multicolumn\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eStudent Data\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"k\"\u003e\\multicolumn\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ec|\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eScores\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Class \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Math \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Science \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII A \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 80 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 78 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Budi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII B \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 85 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 81 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe command \u003ccode\u003e\\multicolumn{3}{|c|}{Student Data}\u003c/code\u003e merges three columns into one centered cell with vertical lines on both sides.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJmSdFa_g7-mgInh_LRdVnNKVRmw9DZ_GCHUW9-Fq42mWjCJ_N-mBDzW1ClMvZK_f31Z1u00StaIEBWm8rAvjYS50IG01Lg93JcNm-0A4OyakBoJNbxdQjYUp6bd5-mPWLNbMdtqyR28I/s1055/tabel4.png' alt=\"Merging columns in LaTeX tables\" \u003e\u003c/p\u003e\n\u003ch3 id=\"merging-columns-and-rows\" class=\"headerLink\"\u003e\n \u003ca href=\"#merging-columns-and-rows\" class=\"header-mark\" aria-label=\"Direct link to section: Merging Columns and Rows\"\u003e\u003c/a\u003eMerging Columns and Rows\u003c/h3\u003e\u003cp\u003eBoth techniques can be combined in the same table.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|c|c|c|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\multirow\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e*\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eNo\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"k\"\u003e\\multirow\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e*\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eName\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"k\"\u003e\\multicolumn\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e2\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ec|\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eScores\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\cline\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3-4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Midterm \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Final \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 80 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 85 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Budi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 78 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 82 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhB5fc9ajN6VMuKFasqhOBa2bWg6UkGFcQFBubVQpP09Hynm_IOKe4oNCFJSUgjQhAGzOmoUhdb1LBtMx7Das5R2F5omEJoTcAvk7RWZw99_lTy5-2u-jyslCw_5NhPvTJtSDPypU-Y1Y8/s1215/tabel5.png' alt=\"Merging rows and columns in LaTeX tables\" \u003e\u003c/p\u003e\n\u003ch3 id=\"rotating-a-table\" class=\"headerLink\"\u003e\n \u003ca href=\"#rotating-a-table\" class=\"header-mark\" aria-label=\"Direct link to section: Rotating a Table\"\u003e\u003c/a\u003eRotating a Table\u003c/h3\u003e\u003cp\u003eIf the document uses landscape orientation or the table is too wide, you can rotate it with the \u003ccode\u003erotating\u003c/code\u003e package.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003erotating\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esidewaystable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|c|c|c|c|c|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Class \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Score 1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Score 2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Score 3 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII A \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 80 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 82 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 84 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esidewaystable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhX9dO4jGFv4L5aTSWofwTMNtQ2rUyIcMriyPBkk3S6DvDOOpbh2RtBSYQRU5J8liBeYgrDinhBBfaNxhZw1XxEe8CNnYnGJ_qXtwsJCF7A3e57VHqgDNkq6bvrB6VBwOKPZzd9-X5t_go/s1047/tabel6.png' alt=\"Landscape table in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eAnother option is the \u003ccode\u003eadjustbox\u003c/code\u003e package.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eadjustbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h!]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eadjustbox\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eangle=90\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|c|c|c|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Class \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Score \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e VII A \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 80 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eadjustbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic2YZeqeVHyBvbKfNiZT7gOLNANnCcSlyltQ41VYZp_5vSFf3DW77zKSW3lIVMQ8UozpnecdY9wT9goJ-minKZ_xVWWl8hDkSipmLLeYrHes-dmZoKb57SRBE8gceCeeNEMhI2wRZy4YM/s1216/tabel7.png' alt=\"Rotated table with adjustbox in LaTeX\" \u003e\u003c/p\u003e\n\u003ch3 id=\"table-position\" class=\"headerLink\"\u003e\n \u003ca href=\"#table-position\" class=\"header-mark\" aria-label=\"Direct link to section: Table Position\"\u003e\u003c/a\u003eTable Position\u003c/h3\u003e\u003cp\u003eTo control table placement, the usual form is:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[h!]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etable\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eCommon placement markers:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003eh\u003c/code\u003e = \u003cem\u003ehere\u003c/em\u003e, place the table as close as possible to the current location,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003et\u003c/code\u003e = \u003cem\u003etop\u003c/em\u003e, place it at the top of the page,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eb\u003c/code\u003e = \u003cem\u003ebottom\u003c/em\u003e, place it at the bottom of the page,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003ep\u003c/code\u003e = \u003cem\u003epage\u003c/em\u003e, place it on a dedicated float page,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e!\u003c/code\u003e = relax some internal placement rules,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eH\u003c/code\u003e = place it exactly here, usually with the \u003ccode\u003efloat\u003c/code\u003e package.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"tables-with-fixed-width\" class=\"headerLink\"\u003e\n \u003ca href=\"#tables-with-fixed-width\" class=\"header-mark\" aria-label=\"Direct link to section: Tables with Fixed Width\"\u003e\u003c/a\u003eTables with Fixed Width\u003c/h3\u003e\u003cp\u003eIf each column needs a fixed width, use the \u003ccode\u003ep{...}\u003c/code\u003e column type.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|p\u003cspan class=\"nb\"\u003e{\u003c/span\u003e3cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e|p\u003cspan class=\"nb\"\u003e{\u003c/span\u003e5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Description \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e A student with a longer description so the content wraps inside the cell. \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Budi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Another example of a fixed-width column. \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeDxz8vyvJiuIPLsWn28TVQytW3ausxkKaCf8cesxIJyRv_MsiM78UrXoDeKzCUTsjdGE2_dwoYyR5brYlVhrAplt-QYao9rkxzV1stXK8lLdDUh8GO-QP74WFZ8T7JRw0k4sUMoWguws/s1229/tabel8.png' alt=\"Fixed column width in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eIf you want to control the total width of the table and let the columns stretch automatically, use \u003ccode\u003etabularx\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabularx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabularx\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|X|X|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Description \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Columns of type X automatically share the remaining horizontal space. \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Budi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e This is useful for tables with longer text content. \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabularx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdp9SaEPdmaU_7NQTLr6qLoUOwC8Zx-tPvZMFOKyDArum1w_2hvTCJTaIgoxveyH3e0wAsEwiJQotOC3XHMhXNaibDLPWdZl09pq8mPGCO8Nsi8qi9fh3AH0luiGlhOdHxz7YdnUnR_mQ/s1222/tabel9.png' alt=\"Table width with tabularx in LaTeX\" \u003e\u003c/p\u003e\n\u003ch3 id=\"coloring-a-table\" class=\"headerLink\"\u003e\n \u003ca href=\"#coloring-a-table\" class=\"header-mark\" aria-label=\"Direct link to section: Coloring a Table\"\u003e\u003c/a\u003eColoring a Table\u003c/h3\u003e\u003cp\u003eTo color table cells or rows, use the \u003ccode\u003excolor\u003c/code\u003e package with the \u003ccode\u003etable\u003c/code\u003e option.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[table]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-16\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e|c|l|c|\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rowcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egray!30\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eNo \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Name \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Score \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e1 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Andi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 80 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e2 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Budi \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 85 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\rowcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eyellow!20\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e3 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e Citra \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e 90 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e \u003cspan class=\"k\"\u003e\\hline\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etabular\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ-JNnp3FYGahUPDofFisapTRKiybHImA8-871IYON5N7nRebAZYjuAM6B5dkaKRGqNBZxp2rQlK-S_oJtkT2ke25pkODnPoHSDHIUs6ntDuQwEBYdbb-JL6LuG30JVTJV1uphKfPZXQY/s1190/tabel10.png' alt=\"Colored table in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eThat concludes this short guide to creating tables in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. You can expand these examples further to match the document you are writing.\u003c/p\u003e\n", "language": "en" }, { "title" : "How to Make a Color Box in LaTeX", "date_published" : "2023-03-01T15:23:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-membuat-color-box-di-latex/", "url" : "https://www.aantriono.com/en/cara-membuat-color-box-di-latex/", "summary": "In many documents, especially math documents, it is useful to place a definition, formula, or note inside a colored frame. This helps the reader focus on important content and makes the page easier to scan.\nHere is one example.\nCreating Color Boxes in LaTeXThe most common package for this purpose is tcolorbox. Add it to the preamble:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eIn many documents, especially math documents, it is useful to place a definition, formula, or note inside a colored frame. This helps the reader focus on important content and makes the page easier to scan.\u003c/p\u003e\n\u003cp\u003eHere is one example.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5OGOA9wFrZSUBCvs0DJOAhAV7KjU3PSsR3BqlZ14_DIap1gAVEXlxEC1_7dNWCzqSA3tQOaQwKzLKBmGJIH6wkPpzIhDg2NQBly2C0XcJw4197DnuV7WmTN0hXCyWKqO7-g71soCs5sQ/s620/colorbox1.png' alt=\"Example of a color box in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"creating-color-boxes-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-color-boxes-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Creating Color Boxes in LaTeX\"\u003e\u003c/a\u003eCreating Color Boxes in LaTeX\u003c/h2\u003e\u003cp\u003eThe most common package for this purpose is \u003ccode\u003etcolorbox\u003c/code\u003e. Add it to the preamble:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eWith \u003ccode\u003etcolorbox\u003c/code\u003e, you can control the frame color, background color, title placement, and many other details. The official documentation is available on CTAN: \u003ccode\u003ehttps://www.ctan.org/pkg/tcolorbox\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eHere are a few simple examples.\u003c/p\u003e\n\u003ch3 id=\"example-1-title-and-body-in-separate-boxes\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1-title-and-body-in-separate-boxes\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1: Title and Body in Separate Boxes\"\u003e\u003c/a\u003eExample 1: Title and Body in Separate Boxes\u003c/h3\u003e\u003cp\u003eIn this example, the title appears like a small label attached to the main content box.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colback=white,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colframe=blue!70!black,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e title=Definition\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eA linear equation in one variable is an equation with exactly one variable\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ewhose highest power is one.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output looks like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcSyPAGES22L3YDgkD-2Ds58O08wYov3ZyJMpGhVvW7w7-vHVtllGIlO3OQ9t3At3NF3fH7SOUcBGwpBSQMXejs7DNVy69pzyqvUxVDAu48QXvsLQSdLwmz-tvrEWh40-epXVgLWlW90U/s559/colorbox3.png' alt=\"Creating colorbox in LaTeX\" \u003e\u003c/p\u003e\n\u003ch3 id=\"example-2-title-in-a-small-box-at-the-top\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2-title-in-a-small-box-at-the-top\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2: Title in a Small Box at the Top\"\u003e\u003c/a\u003eExample 2: Title in a Small Box at the Top\u003c/h3\u003e\u003cp\u003eYou can also emphasize the title by placing it in a small box above the main body.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e enhanced,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colback=yellow!10,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colframe=red!70!black,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e coltitle=white,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e boxed title style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecolback=red!70!black\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e title=Definition\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eLet \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ea\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e, \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003eb\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e, and \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ec\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e be real numbers with \u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nb\"\u003ea \u003c/span\u003e\u003cspan class=\"nv\"\u003e\\ne\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eA quadratic equation has the general form\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"sb\"\u003e\\[\u003c/span\u003e\u003cspan class=\"nb\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003eax^\u003c/span\u003e\u003cspan class=\"m\"\u003e2\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003e bx \u003c/span\u003e\u003cspan class=\"o\"\u003e+\u003c/span\u003e\u003cspan class=\"nb\"\u003e c \u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"nb\"\u003e \u003c/span\u003e\u003cspan class=\"m\"\u003e0\u003c/span\u003e\u003cspan class=\"nb\"\u003e.\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e\u003c/span\u003e\u003cspan class=\"s\"\u003e\\]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output looks like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0nZSFUldtzascYOEZ1uqawv4k9AeDvTF1lBkex8kuvKkCI4LTO4fgeIXOhfEva-LBZxt0ml4wTBe_pAfrA-mhsA7SPamGIEs85n7rKdk3IrApGp8DW0c7l9QdrApvLPMtev23MBxY_H8/s620/colorbox1.png' alt=\"Creating colorbox in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eWhat if you want a more decorative box like the following?\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglaEZbG93PPPpiNchSzpIh7_PnF4YQY6033gDGPkGVBB657w10HaNbimGd1_KBMaOifUN5YGXYZmCZ2k7MkjZ6wpeet5GsjhO9PbvC5yLCgLqquuwZPB2S5_FVRtWV2XimIzKoucmvGHQ/s536/colorbox2.png' alt=\"Example of a color box with bclogo\" \u003e\u003c/p\u003e\n\u003ch3 id=\"example-3-using-the-bclogo-package\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3-using-the-bclogo-package\" class=\"header-mark\" aria-label=\"Direct link to section: Example 3: Using the bclogo Package\"\u003e\u003c/a\u003eExample 3: Using the \u003ccode\u003ebclogo\u003c/code\u003e Package\u003c/h3\u003e\u003cp\u003eFor that style, you can use the \u003ccode\u003ebclogo\u003c/code\u003e package.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebclogo\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen create the box like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebclogo\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[couleur=blue!15, arrondi=0.1, logo=\u003cspan class=\"k\"\u003e\\bcinfo\u003c/span\u003e]\u003cspan class=\"nb\"\u003e{\u003c/span\u003eNote\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eIf two parallel lines are cut by another line, then corresponding angles\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ehave the same measure.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebclogo\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThose examples show the basic idea of building color boxes in LaTeX. You can extend them further with different colors, icons, border styles, and title layouts.\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat is a short guide to creating color boxes in LaTeX. This feature is especially useful when writing worksheets, lecture notes, or math handouts that need clear visual emphasis.\u003c/p\u003e\n", "language": "en" }, { "title" : "LaTeX Document Structure", "date_published" : "2023-03-01T14:03:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/struktur-dokumen-latex-2023-03/", "url" : "https://www.aantriono.com/en/struktur-dokumen-latex-2023-03/", "summary": "LaTeX has a document structure organized through document classes and supporting packages. These packages control many aspects of a document, including font size, page layout, table of contents, chapters, subsections, images, and text alignment. Because of that, choosing the right packages is important for improving the quality of your LaTeX\\LaTeXLATE​X document.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eLaTeX has a document structure organized through document classes and supporting packages. These packages control many aspects of a document, including font size, page layout, table of contents, chapters, subsections, images, and text alignment. Because of that, choosing the right packages is important for improving the quality of your \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document.\u003c/p\u003e\n\u003ch2 id=\"basic-structure-of-latex-documents\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-structure-of-latex-documents\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Structure of LaTeX Documents\"\u003e\u003c/a\u003eBasic Structure of LaTeX Documents\u003c/h2\u003e\u003cp\u003eA \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document has a basic structure marked by blocks enclosed by pairs of \u003ccode\u003e\\begin\u003c/code\u003e and \u003ccode\u003e\\end\u003c/code\u003e commands. A simple example is shown below:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 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9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eHello, world!\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eEach command in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e begins with a \u003cem\u003ebackslash\u003c/em\u003e (\u003ccode\u003e\\\u003c/code\u003e). In addition, every document should start with the command \u003ccode\u003e\\documentclass{...}\u003c/code\u003e to define the type of document being processed.\u003c/p\u003e\n\u003ch3 id=\"document-class-type\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-class-type\" class=\"header-mark\" aria-label=\"Direct link to section: Document Class Type\"\u003e\u003c/a\u003eDocument Class Type\u003c/h3\u003e\u003cp\u003eThere are several document classes in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, including the following.\u003c/p\u003e\n\u003ch4 id=\"general-document-classes\" class=\"headerLink\"\u003e\n \u003ca href=\"#general-document-classes\" class=\"header-mark\" aria-label=\"Direct link to section: General Document Classes\"\u003e\u003c/a\u003eGeneral Document Classes\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003e\u003cem\u003earticle\u003c/em\u003e. The most common class for writing journal articles, academic papers, worksheets, and other documents without chapters.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003ereport\u003c/em\u003e. Used for longer documents that contain chapters, sections, and subsections.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003ebook\u003c/em\u003e. Used for writing books that consist of multiple chapters.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"special-document-classes\" class=\"headerLink\"\u003e\n \u003ca href=\"#special-document-classes\" class=\"header-mark\" aria-label=\"Direct link to section: Special Document Classes\"\u003e\u003c/a\u003eSpecial Document Classes\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003e\u003cem\u003eletter\u003c/em\u003e. Used for official letters.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003ebeamer\u003c/em\u003e. Used for presentation documents.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003eslides\u003c/em\u003e. Used for presentations with large sans serif text.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"packagepackage\" class=\"headerLink\"\u003e\n \u003ca href=\"#packagepackage\" class=\"header-mark\" aria-label=\"Direct link to section: Package/Package\"\u003e\u003c/a\u003ePackage/Package\u003c/h3\u003e\u003cp\u003eThe part between \u003ccode\u003e\\documentclass[...] {...}\u003c/code\u003e and \u003ccode\u003e\\begin{document}\u003c/code\u003e is called the \u003cem\u003epreamble\u003c/em\u003e. In this section, you can declare the packages needed for your document. For example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[12pt,a4paper]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[english]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eamsmath,amssymb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebackground\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eInformation:\u003c/p\u003e\n\u003ch4 id=\"packages-for-layout-and-language\" class=\"headerLink\"\u003e\n \u003ca href=\"#packages-for-layout-and-language\" class=\"header-mark\" aria-label=\"Direct link to section: Packages for Layout and Language\"\u003e\u003c/a\u003ePackages for Layout and Language\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003e\u003cem\u003ebabel\u003c/em\u003e. Used to set language rules in documents.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003egeometry\u003c/em\u003e. Used to define the layout of the document.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"package-for-mathematics\" class=\"headerLink\"\u003e\n \u003ca href=\"#package-for-mathematics\" class=\"header-mark\" aria-label=\"Direct link to section: Package for Mathematics\"\u003e\u003c/a\u003ePackage for Mathematics\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003e\u003cem\u003eamsmath\u003c/em\u003e. Used for writing mathematics with more complete features.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003eamsymb\u003c/em\u003e. Used for mathematical symbols.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"package-for-multimedia-and-display\" class=\"headerLink\"\u003e\n \u003ca href=\"#package-for-multimedia-and-display\" class=\"header-mark\" aria-label=\"Direct link to section: Package for Multimedia and Display\"\u003e\u003c/a\u003ePackage for Multimedia and Display\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003e\u003cem\u003egraphicx\u003c/em\u003e. Used to insert external images.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003excolor\u003c/em\u003e. Used to apply color in documents.\u003c/li\u003e\n\u003cli\u003e\u003cem\u003ebackground\u003c/em\u003e. Used to add a document background.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"document-body-and-examples\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-body-and-examples\" class=\"header-mark\" aria-label=\"Direct link to section: Document Body and Examples\"\u003e\u003c/a\u003eDocument Body and Examples\u003c/h3\u003e\u003cp\u003eThe part between \u003ccode\u003e\\begin{document}\u003c/code\u003e and \u003ccode\u003e\\end{document}\u003c/code\u003e is called the \u003cem\u003edocument body\u003c/em\u003e. In this section, you can write plain text as well as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e commands. For practice, copy the following code into your \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e editor, either online or offline.\u003c/p\u003e\n\u003ch4 id=\"latex-code-example\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-code-example\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Code Example\"\u003e\u003c/a\u003eLaTeX Code Example\u003c/h4\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[12pt,a4paper]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[english]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSample Document\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis is a sample \u003cspan class=\"k\"\u003e\\LaTeX\\ \u003c/span\u003edocument.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eThis text is blue.\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch4 id=\"compilation-results\" class=\"headerLink\"\u003e\n \u003ca href=\"#compilation-results\" class=\"header-mark\" aria-label=\"Direct link to section: Compilation Results\"\u003e\u003c/a\u003eCompilation Results\u003c/h4\u003e\u003cp\u003eThe compiled result will look like the following.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjLoaEAjskrvaauDivkKLilDlOG64PFkNSSTo5EmYfVP66NpjjkaOQIgqKBytFBKoX6dYPwEsCaI9Z_tuDwMGkew464MHo3r0QcMaTJfBHc-F8rOHmmZ4L91q98-dfF2yuXwEzfPyFanw/s1186/Screenshot_3.png' alt=\"Sample compiled output of a LaTeX document structure\" \u003e\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat concludes this overview of basic \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document structure. I hope it provides a practical starting point for writing your own documents.\u003c/p\u003e\n", "language": "en" }, { "title" : "LaTeX Installation", "date_published" : "2023-03-01T14:03:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pemasangan-latex/", "url" : "https://www.aantriono.com/en/pemasangan-latex/", "summary": "Installing LaTeX\\LaTeXLATE​X basically means preparing the tools needed to run the TeX\\TeXTE​X system. To start creating LaTeX\\LaTeXLATE​X documents, we usually need:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eInstalling \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e basically means preparing the tools needed to run the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e system. To start creating \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e documents, we usually need:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ea \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e distribution, such as Mik\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e Live, or MacTeX;\u003c/li\u003e\n\u003cli\u003ea document viewer, such as Adobe Reader, Foxit Reader, or Sumatra PDF;\u003c/li\u003e\n\u003cli\u003ea \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e editor, such as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eworks or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003estudio.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"two-ways-to-use-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#two-ways-to-use-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Two Ways to Use LaTeX\"\u003e\u003c/a\u003eTwo Ways to Use LaTeX\u003c/h2\u003e\u003cp\u003eAfter those programs are installed, you can start using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. In general, there are two ways to work with it: online and offline.\u003c/p\u003e\n\u003ch3 id=\"1-online-mode\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-online-mode\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Online Mode\"\u003e\u003c/a\u003e1. Online Mode\u003c/h3\u003e\u003cp\u003eWhen using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e online, the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e system is already installed on the service provider\u0026rsquo;s server, usually together with an editor. Some services you can use are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ca href=\"http://www.overleaf.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eOverleaf\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"http://www.papeeria.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ePapeeria\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eTo use those services, you need to create an account first.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWEcWNStn439b0IvhqOvubX7d9XpXQuPNwMOBtRaqVi3juK11ZOTM4eDW8Hb_tt472lMBC5lVxbi_Go3ojkUaXAEopFOWAtkh0Apn9xV2JPf07ZoLEwbNjpCQEo7RbznDvGk8VZlO-7QI/s1181/Screenshot_1.png' alt=\"Overleaf.com\" \u003e\u003c/p\u003e\n\u003ch3 id=\"2-offline-mode\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-offline-mode\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Offline Mode\"\u003e\u003c/a\u003e2. Offline Mode\u003c/h3\u003e\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is available on different operating systems with different distributions:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eWindows\u003c/strong\u003e: Mik\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e Live, Pro\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, PC\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and BaKom\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMac\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e Live and Mac\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLinux\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e Live\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"latex-editors-for-offline-mode\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-editors-for-offline-mode\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Editors for Offline Mode\"\u003e\u003c/a\u003eLaTeX Editors for Offline Mode\u003c/h3\u003e\u003cp\u003eSome editors that can be used for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are TeXmaker, LyX, TeXworks, TeXstudio, and Gummi.\u003c/p\u003e\n\u003cp\u003eDownload links:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eMik\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e: \u003ca href=\"https://miktex.org/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003emiktex.org\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e Live: \u003ca href=\"https://www.tug.org/texlive/acquire-iso.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003etug.org/texlive/acquire-iso.html\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eTeXstudio: \u003ca href=\"https://texstudio.org/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003etexstudio.org\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eChoose the distribution and editor that match your operating system, then follow the installation instructions carefully.\u003c/p\u003e\n\u003ch2 id=\"a-simple-latex-test\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-simple-latex-test\" class=\"header-mark\" aria-label=\"Direct link to section: A Simple LaTeX Test\"\u003e\u003c/a\u003eA Simple LaTeX Test\u003c/h2\u003e\u003cp\u003eAfter installation, try writing the following simple document in your \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e editor, then compile it and see the result.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eHello, world!\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eThis is my first document written in \u003cspan class=\"k\"\u003e\\LaTeX\u003c/span\u003e.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you run the code in Overleaf, the result will look like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEzmIx6xUXtl1NU1lr5raF5UINPTHn0Pnx78pyaG8nFvIt8ZSroJkd42RfFYjlYC2w2nHPfW2tzR8valm7Co-YdFz1DTHO7HY_wRml0BCSO6wCvMaqaB9gy13KcrUvuiPAFEbGftaYUVo/s1191/Screenshot_2.png' alt=\"LaTeX document in Overleaf\" \u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eOnce the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e system, editor, and PDF viewer are installed, you are ready to start writing documents. The next step is to learn document structure, packages, and the most common commands.\u003c/p\u003e\n", "language": "en" }, { "title" : "Drawing a Rectangle", "date_published" : "2023-03-01T14:02:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-persegi-panjang/", "url" : "https://www.aantriono.com/en/menggambar-persegi-panjang/", "summary": " Drawing and Animating Rectangles in GeoGebraTo draw and animate a rectangle in GeoGebra, follow the steps below.\nStep 1: Creating a SliderCreate a slider by clicking the Slider tool, then click the GeoGebra workspace. The following dialog box will appear.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-and-animating-rectangles-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-and-animating-rectangles-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing and Animating Rectangles in GeoGebra\"\u003e\u003c/a\u003eDrawing and Animating Rectangles in GeoGebra\u003c/h2\u003e\u003cp\u003eTo draw and animate a rectangle in GeoGebra, follow the steps below.\u003c/p\u003e\n\u003ch3 id=\"step-1-creating-a-slider\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-1-creating-a-slider\" class=\"header-mark\" aria-label=\"Direct link to section: Step 1: Creating a Slider\"\u003e\u003c/a\u003eStep 1: Creating a Slider\u003c/h3\u003e\u003cp\u003eCreate a slider by clicking the \u003cstrong\u003eSlider\u003c/strong\u003e tool, then click the GeoGebra workspace. The following dialog box will appear.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhipUnJqSzFHigigPoAaWrFuJg69Nd2M-orbkpVw4KOvizlx5l5BsGT1MeHr73smre7mpiGCbq8dqzAdfegYKCjaapC8S1zeSxcoAGUON9uA6YGBUF1gh5tNbXJHm53J6DZma0It4BKEbw/s1053/Screenshot_4.png\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\n \u0026lt;img border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;568\u0026quot; data-original-width=\u0026quot;1053\u0026quot; height=\u0026quot;216\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhipUnJqSzFHigigPoAaWrFuJg69Nd2M-orbkpVw4KOvizlx5l5BsGT1MeHr73smre7mpiGCbq8dqzAdfegYKCjaapC8S1zeSxcoAGUON9uA6YGBUF1gh5tNbXJHm53J6DZma0It4BKEbw/w400-h216/Screenshot_4.png\u0026quot; title=\u0026quot;slider geogebra\u0026quot; width=\u0026quot;400\u0026quot; style=\u0026quot;max-width: 100%; height: auto;\u0026quot; alt=\u0026quot;Dialog pembuatan slider di GeoGebra\u0026quot; /\u0026gt;\n\u0026lt;/a\u0026gt;\n\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eFill in the dialog box above: \u003ccode\u003eName\u003c/code\u003e for the slider name, \u003ccode\u003eMin\u003c/code\u003e for the minimum value, \u003ccode\u003eMax\u003c/code\u003e for the maximum value, and \u003ccode\u003eIncrement\u003c/code\u003e for the step size. Then click \u003ccode\u003eOK\u003c/code\u003e and the slider will appear. Use the \u003cstrong\u003eMove\u003c/strong\u003e tool to drag the slider.\u003c/p\u003e\n\u003ch3 id=\"step-2-create-a-rectangle-with-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-2-create-a-rectangle-with-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Step 2: Create a Rectangle with Lines\"\u003e\u003c/a\u003eStep 2: Create a Rectangle with Lines\u003c/h3\u003e\u003cp\u003eTo create the rectangle, enter the line equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey=a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e in the \u003cstrong\u003eInput\u003c/strong\u003e box on the left. In the same way, enter \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=6a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey=4a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. After that, the display will look like the image below.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX3hOHwsfEA7_eYli0mu2lehd9rAR1ab3jm1SnHOGfMOz4_gFm2N1sedXNELJGgi6vsWUQwAafD5szGoS4xXydBBKpnOMvsqWrIAkyCTPdhyphenhyphenLI8Ymm_BZn4_wLY1QzbDOVNgWnZsVG3-8/s1289/Screenshot_5.png\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\n \u0026lt;img border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;547\u0026quot; data-original-width=\u0026quot;1289\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX3hOHwsfEA7_eYli0mu2lehd9rAR1ab3jm1SnHOGfMOz4_gFm2N1sedXNELJGgi6vsWUQwAafD5szGoS4xXydBBKpnOMvsqWrIAkyCTPdhyphenhyphenLI8Ymm_BZn4_wLY1QzbDOVNgWnZsVG3-8/s320/Screenshot_5.png\u0026quot; width=\u0026quot;320\u0026quot; style=\u0026quot;max-width: 100%; height: auto;\u0026quot; alt=\u0026quot;Garis-garis pembentuk persegi panjang di GeoGebra\u0026quot; /\u0026gt;\n\u0026lt;/a\u0026gt;\n\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"step-3-creating-cutting-points\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-3-creating-cutting-points\" class=\"header-mark\" aria-label=\"Direct link to section: Step 3: Creating Cutting Points\"\u003e\u003c/a\u003eStep 3: Creating Cutting Points\u003c/h3\u003e\u003cp\u003eCreate the intersection points \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e by clicking the \u003cstrong\u003eIntersect\u003c/strong\u003e tool on the toolbar, then selecting the two lines that form each point.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYCJQ-cgZQ0VOqLw2-LRU3SBvqacMLDCzObcXX4lhZsxZ3AHfiYVRsjNQleJTjXQP3-UcF5a0yZY6H4Y6Sq1y5agOq-Tr_4-Nhb1yrRu1FkxvIq_1Efh7yoCaE4XE5Aq6n3VhrYU4o3Ic/s1290/Screenshot_6.png\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\n \u0026lt;img border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;527\u0026quot; data-original-width=\u0026quot;1290\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYCJQ-cgZQ0VOqLw2-LRU3SBvqacMLDCzObcXX4lhZsxZ3AHfiYVRsjNQleJTjXQP3-UcF5a0yZY6H4Y6Sq1y5agOq-Tr_4-Nhb1yrRu1FkxvIq_1Efh7yoCaE4XE5Aq6n3VhrYU4o3Ic/s320/Screenshot_6.png\u0026quot; width=\u0026quot;320\u0026quot; style=\u0026quot;max-width: 100%; height: auto;\u0026quot; alt=\u0026quot;Titik potong pada garis di GeoGebra\u0026quot; /\u0026gt;\n\u0026lt;/a\u0026gt;\n\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"step-4-creating-rectangular-polygons\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-4-creating-rectangular-polygons\" class=\"header-mark\" aria-label=\"Direct link to section: Step 4: Creating Rectangular Polygons\"\u003e\u003c/a\u003eStep 4: Creating Rectangular Polygons\u003c/h3\u003e\u003cp\u003eCreate the rectangle by connecting the intersection points. Click the \u003cstrong\u003ePolygon\u003c/strong\u003e tool on the toolbar, then click points \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and finally return to point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikZBS8TQYNKzGBO3nQSP46jBEQoPspnklTx2zLCehjlPheHxjKesf9ZHZ9NeGGp4ffokOG5KkWJ7YDV31Dx9_ioIXe_SWdurfwK8qSCingWxpxTDJiWsM-w0ibfP8jvmuS_dX6CkSfb5c/s1291/Screenshot_7.png\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\n \u0026lt;img border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;547\u0026quot; data-original-width=\u0026quot;1291\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikZBS8TQYNKzGBO3nQSP46jBEQoPspnklTx2zLCehjlPheHxjKesf9ZHZ9NeGGp4ffokOG5KkWJ7YDV31Dx9_ioIXe_SWdurfwK8qSCingWxpxTDJiWsM-w0ibfP8jvmuS_dX6CkSfb5c/s320/Screenshot_7.png\u0026quot; width=\u0026quot;320\u0026quot; style=\u0026quot;max-width: 100%; height: auto;\u0026quot; alt=\u0026quot;Persegi panjang yang terbentuk di GeoGebra\u0026quot; /\u0026gt;\n\u0026lt;/a\u0026gt;\n\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"step-5-hiding-the-auxiliary-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-5-hiding-the-auxiliary-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Step 5: Hiding the Auxiliary Lines\"\u003e\u003c/a\u003eStep 5: Hiding the Auxiliary Lines\u003c/h3\u003e\u003cp\u003eHide the auxiliary lines created earlier by right-clicking each line and disabling \u003cstrong\u003eShow Object\u003c/strong\u003e.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZsvmVqCeGtGlzVLrwAZ4YZL9azTXEmHDCu67pJOYPejCZMqnGyfdFDpEXSTVn26AOLeJk6OaodWQn_HSLEbDPgKfE6lpN5DB7r-QeuS5cApm_5BAb037cHpI2c7Q0rOPHHvoJqt_7uSw/s1289/Screenshot_8.png\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\n \u0026lt;img border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;578\u0026quot; data-original-width=\u0026quot;1289\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZsvmVqCeGtGlzVLrwAZ4YZL9azTXEmHDCu67pJOYPejCZMqnGyfdFDpEXSTVn26AOLeJk6OaodWQn_HSLEbDPgKfE6lpN5DB7r-QeuS5cApm_5BAb037cHpI2c7Q0rOPHHvoJqt_7uSw/s320/Screenshot_8.png\u0026quot; width=\u0026quot;320\u0026quot; style=\u0026quot;max-width: 100%; height: auto;\u0026quot; alt=\u0026quot;Garis bantu disembunyikan di GeoGebra\u0026quot; /\u0026gt;\n\u0026lt;/a\u0026gt;\n\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"step-6-testing-animation-with-sliders\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-6-testing-animation-with-sliders\" class=\"header-mark\" aria-label=\"Direct link to section: Step 6: Testing Animation with Sliders\"\u003e\u003c/a\u003eStep 6: Testing Animation with Sliders\u003c/h3\u003e\u003cp\u003eWhen the side slider is moved, the area and perimeter of the rectangle change. This shows that the side length affects both area and perimeter.\u003c/p\u003e\n\u003ch3 id=\"step-7-tidying-up-the-appearance\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-7-tidying-up-the-appearance\" class=\"header-mark\" aria-label=\"Direct link to section: Step 7: Tidying Up the Appearance\"\u003e\u003c/a\u003eStep 7: Tidying Up the Appearance\u003c/h3\u003e\u003cp\u003eTidy up the display by hiding the coordinate axes. To change the color of the slider and rectangle, right-click each object, open the settings, and choose a color.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUIS2-wmR41k9zT2FEpOlXvg2sgP1ou7bHtxFNo4eFKrrpPwmTlNiSyroxBLN8CJwjYnsfo-KU8Ovr_FF8acCEcCl9Hv7yys2EKNiMAdW8Qg_sQ_7eFH42wX7FhIS-4t7fbTk32ipDTt4/s1290/Screenshot_9.png\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\n \u0026lt;img border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;566\u0026quot; data-original-width=\u0026quot;1290\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUIS2-wmR41k9zT2FEpOlXvg2sgP1ou7bHtxFNo4eFKrrpPwmTlNiSyroxBLN8CJwjYnsfo-KU8Ovr_FF8acCEcCl9Hv7yys2EKNiMAdW8Qg_sQ_7eFH42wX7FhIS-4t7fbTk32ipDTt4/s320/Screenshot_9.png\u0026quot; width=\u0026quot;320\u0026quot; style=\u0026quot;max-width: 100%; height: auto;\u0026quot; alt=\u0026quot;Tampilan akhir persegi panjang di GeoGebra\u0026quot; /\u0026gt;\n\u0026lt;/a\u0026gt;\n\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"video-tutorials\" class=\"headerLink\"\u003e\n \u003ca href=\"#video-tutorials\" class=\"header-mark\" aria-label=\"Direct link to section: Video Tutorials\"\u003e\u003c/a\u003eVideo Tutorials\u003c/h3\u003e\u003cp\u003eThe following video tutorial may help clarify the steps above.\u003c/p\u003e\n\u003ciframe allowfullscreen=\"\" class=\"BLOG_video_class\" height=\"266\" src=\"https://www.youtube.com/embed/EiAzGa0JuAU\" width=\"320\" youtube-src-id=\"EiAzGa0JuAU\" style=\"max-width: 100%;\"\u003e\u003c/iframe\u003e\n\u003ch3 id=\"example-of-final-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-of-final-result\" class=\"header-mark\" aria-label=\"Direct link to section: Example of Final Result\"\u003e\u003c/a\u003eExample of Final Result\u003c/h3\u003e\u003cp\u003eBelow is an example of the final rectangle animation. Move the red slider \u003ccode\u003ea\u003c/code\u003e to the right.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/kavm95cf/width/600/height/400/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; max-width: 100%;\" title=\"Animasi Persegi Panjang\" width=\"600px\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat is the guide to drawing and animating rectangles in GeoGebra. If you have suggestions or feedback, please leave a comment or contact me through the contact menu. I hope this is useful.\u003c/p\u003e\n", "language": "en" }, { "title" : "How to Write LaTeX Using Overleaf", "date_published" : "2023-03-01T14:01:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-menulis-latex-menggunakan-overleaf/", "url" : "https://www.aantriono.com/en/cara-menulis-latex-menggunakan-overleaf/", "summary": " Writing LaTeX with Overleaf.comOne of the easiest ways to write LaTeX\\LaTeXLATE​X online is to use Overleaf. This guide shows the basic flow: create an account, start your first project, and compile your document.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"writing-latex-with-overleafcom\" class=\"headerLink\"\u003e\n \u003ca href=\"#writing-latex-with-overleafcom\" class=\"header-mark\" aria-label=\"Direct link to section: Writing LaTeX with Overleaf.com\"\u003e\u003c/a\u003eWriting LaTeX with Overleaf.com\u003c/h2\u003e\u003cp\u003eOne of the easiest ways to write \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e online is to use \u003ca href=\"https://www.overleaf.com/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eOverleaf\u003c/a\u003e. This guide shows the basic flow: create an account, start your first project, and compile your document.\u003c/p\u003e\n\u003ch3 id=\"registration-and-initial-setup\" class=\"headerLink\"\u003e\n \u003ca href=\"#registration-and-initial-setup\" class=\"header-mark\" aria-label=\"Direct link to section: Registration and Initial Setup\"\u003e\u003c/a\u003eRegistration and Initial Setup\u003c/h3\u003e\u003ch4 id=\"1-open-the-overleaf-website\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-open-the-overleaf-website\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Open the Overleaf website\"\u003e\u003c/a\u003e1. Open the Overleaf website\u003c/h4\u003e\u003cp\u003eVisit \u003ca href=\"https://www.overleaf.com/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.overleaf.com/\u003c/a\u003e.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYkzxzg4TlYYdydq0HiZq9urnMfXeDy8uiZJF7YEW-blb1KyScHHYblYdTiFJSbzmPRIsRpkKyuKlG8xb5xRJnlbtqUZryT5SUGe7COQC59UIKNnXqXwGXC1PHvddR_gBaGTlFIpDpqCE/w320-h152/overleaf1.png\" title=\"Overleaf homepage\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYkzxzg4TlYYdydq0HiZq9urnMfXeDy8uiZJF7YEW-blb1KyScHHYblYdTiFJSbzmPRIsRpkKyuKlG8xb5xRJnlbtqUZryT5SUGe7COQC59UIKNnXqXwGXC1PHvddR_gBaGTlFIpDpqCE/w320-h152/overleaf1.png\" data-sub-html=\"\u003ch2\u003eOverleaf homepage\u003c/h2\u003e\u003cp\u003eOverleaf homepage\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYkzxzg4TlYYdydq0HiZq9urnMfXeDy8uiZJF7YEW-blb1KyScHHYblYdTiFJSbzmPRIsRpkKyuKlG8xb5xRJnlbtqUZryT5SUGe7COQC59UIKNnXqXwGXC1PHvddR_gBaGTlFIpDpqCE/w320-h152/overleaf1.png' alt=\"Overleaf homepage\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eOverleaf homepage\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYkzxzg4TlYYdydq0HiZq9urnMfXeDy8uiZJF7YEW-blb1KyScHHYblYdTiFJSbzmPRIsRpkKyuKlG8xb5xRJnlbtqUZryT5SUGe7COQC59UIKNnXqXwGXC1PHvddR_gBaGTlFIpDpqCE/s1106/overleaf1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYkzxzg4TlYYdydq0HiZq9urnMfXeDy8uiZJF7YEW-blb1KyScHHYblYdTiFJSbzmPRIsRpkKyuKlG8xb5xRJnlbtqUZryT5SUGe7COQC59UIKNnXqXwGXC1PHvddR_gBaGTlFIpDpqCE/s1106/overleaf1.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch4 id=\"2-click-register\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-click-register\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Click Register\"\u003e\u003c/a\u003e2. Click \u003ccode\u003eRegister\u003c/code\u003e\u003c/h4\u003e\u003cp\u003eYou will be taken to the registration page.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifZt89STrkznStIWMA1L3XptDORu_f1jxoubJskLRXlgC-FSTS5KJsUjxYhmjCx2NGsxWVyd0CiJDPqC733My7jK-ui3KAq1FjwQ2NWAZb-gdK5m5YCkjv7bYrXvuGOMBZmqsNT9IDYmo/w320-h166/overleaf2.png\" title=\"Overleaf registration form\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifZt89STrkznStIWMA1L3XptDORu_f1jxoubJskLRXlgC-FSTS5KJsUjxYhmjCx2NGsxWVyd0CiJDPqC733My7jK-ui3KAq1FjwQ2NWAZb-gdK5m5YCkjv7bYrXvuGOMBZmqsNT9IDYmo/w320-h166/overleaf2.png\" data-sub-html=\"\u003ch2\u003eOverleaf registration form\u003c/h2\u003e\u003cp\u003eOverleaf registration form\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifZt89STrkznStIWMA1L3XptDORu_f1jxoubJskLRXlgC-FSTS5KJsUjxYhmjCx2NGsxWVyd0CiJDPqC733My7jK-ui3KAq1FjwQ2NWAZb-gdK5m5YCkjv7bYrXvuGOMBZmqsNT9IDYmo/w320-h166/overleaf2.png' alt=\"Overleaf registration form\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eOverleaf registration form\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifZt89STrkznStIWMA1L3XptDORu_f1jxoubJskLRXlgC-FSTS5KJsUjxYhmjCx2NGsxWVyd0CiJDPqC733My7jK-ui3KAq1FjwQ2NWAZb-gdK5m5YCkjv7bYrXvuGOMBZmqsNT9IDYmo/s1108/overleaf2.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifZt89STrkznStIWMA1L3XptDORu_f1jxoubJskLRXlgC-FSTS5KJsUjxYhmjCx2NGsxWVyd0CiJDPqC733My7jK-ui3KAq1FjwQ2NWAZb-gdK5m5YCkjv7bYrXvuGOMBZmqsNT9IDYmo/s1108/overleaf2.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eEnter your email address and password, then click \u003ccode\u003eRegister using your email\u003c/code\u003e. After that, confirm the account from the verification email.\u003c/p\u003e\n\u003ch3 id=\"creating-your-first-project\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-your-first-project\" class=\"header-mark\" aria-label=\"Direct link to section: Creating Your First Project\"\u003e\u003c/a\u003eCreating Your First Project\u003c/h3\u003e\u003ch4 id=\"3-click-create-first-project\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-click-create-first-project\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Click Create First Project\"\u003e\u003c/a\u003e3. Click \u003ccode\u003eCreate First Project\u003c/code\u003e\u003c/h4\u003e\u003cp\u003eAfter signing in, Overleaf will prompt you to create your first project.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPH8GqGRrXzuGrl-dHphWwRqqc4XcfQVFqO8FdkDbjZhuY1VUlQXmmxw-lkkNLC_LJASNiWTNcIyanDKAlJnJ3HHuMoovtGQP9pjAPN-XKYnScuH_adY-WxQeCrwAFBxH6MDF9ITEvrtk/w320-h96/overleaf3.png\" title=\"Create first project in Overleaf\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPH8GqGRrXzuGrl-dHphWwRqqc4XcfQVFqO8FdkDbjZhuY1VUlQXmmxw-lkkNLC_LJASNiWTNcIyanDKAlJnJ3HHuMoovtGQP9pjAPN-XKYnScuH_adY-WxQeCrwAFBxH6MDF9ITEvrtk/w320-h96/overleaf3.png\" data-sub-html=\"\u003ch2\u003eCreate first project in Overleaf\u003c/h2\u003e\u003cp\u003eCreate first project in Overleaf\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPH8GqGRrXzuGrl-dHphWwRqqc4XcfQVFqO8FdkDbjZhuY1VUlQXmmxw-lkkNLC_LJASNiWTNcIyanDKAlJnJ3HHuMoovtGQP9pjAPN-XKYnScuH_adY-WxQeCrwAFBxH6MDF9ITEvrtk/w320-h96/overleaf3.png' alt=\"Create first project in Overleaf\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCreate first project in Overleaf\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPH8GqGRrXzuGrl-dHphWwRqqc4XcfQVFqO8FdkDbjZhuY1VUlQXmmxw-lkkNLC_LJASNiWTNcIyanDKAlJnJ3HHuMoovtGQP9pjAPN-XKYnScuH_adY-WxQeCrwAFBxH6MDF9ITEvrtk/s1121/overleaf3.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPH8GqGRrXzuGrl-dHphWwRqqc4XcfQVFqO8FdkDbjZhuY1VUlQXmmxw-lkkNLC_LJASNiWTNcIyanDKAlJnJ3HHuMoovtGQP9pjAPN-XKYnScuH_adY-WxQeCrwAFBxH6MDF9ITEvrtk/s1121/overleaf3.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch4 id=\"4-choose-blank-project\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-choose-blank-project\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Choose Blank Project\"\u003e\u003c/a\u003e4. Choose \u003ccode\u003eBlank Project\u003c/code\u003e\u003c/h4\u003e\u003cp\u003eFor a fresh start, select the blank project option.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYXHTsBfqRzIDFMbYN-oLENcyjGQW8SueJS1zeDMUXN11Kuf2-73kRirtGw7KxTg7RcOm_mp4hM5HwWzYPdMaNDH5TuG7PwdD-5DoghyPTDrTab6PQGEv4jtC7IA66LvO80Jc2qOXBjaU/w320-h149/overleaf4.png\" title=\"Blank Project option in Overleaf\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYXHTsBfqRzIDFMbYN-oLENcyjGQW8SueJS1zeDMUXN11Kuf2-73kRirtGw7KxTg7RcOm_mp4hM5HwWzYPdMaNDH5TuG7PwdD-5DoghyPTDrTab6PQGEv4jtC7IA66LvO80Jc2qOXBjaU/w320-h149/overleaf4.png\" data-sub-html=\"\u003ch2\u003eBlank Project option in Overleaf\u003c/h2\u003e\u003cp\u003eBlank Project option in Overleaf\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYXHTsBfqRzIDFMbYN-oLENcyjGQW8SueJS1zeDMUXN11Kuf2-73kRirtGw7KxTg7RcOm_mp4hM5HwWzYPdMaNDH5TuG7PwdD-5DoghyPTDrTab6PQGEv4jtC7IA66LvO80Jc2qOXBjaU/w320-h149/overleaf4.png' alt=\"Blank Project option in Overleaf\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eBlank Project option in Overleaf\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYXHTsBfqRzIDFMbYN-oLENcyjGQW8SueJS1zeDMUXN11Kuf2-73kRirtGw7KxTg7RcOm_mp4hM5HwWzYPdMaNDH5TuG7PwdD-5DoghyPTDrTab6PQGEv4jtC7IA66LvO80Jc2qOXBjaU/s1111/overleaf4.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYXHTsBfqRzIDFMbYN-oLENcyjGQW8SueJS1zeDMUXN11Kuf2-73kRirtGw7KxTg7RcOm_mp4hM5HwWzYPdMaNDH5TuG7PwdD-5DoghyPTDrTab6PQGEv4jtC7IA66LvO80Jc2qOXBjaU/s1111/overleaf4.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch4 id=\"5-name-the-project\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-name-the-project\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Name the project\"\u003e\u003c/a\u003e5. Name the project\u003c/h4\u003e\u003cp\u003eType a project name that matches your document or task.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis95v1JBh55GhkTflZqjpPmi1J8ECZgB7yXamL8LVsEIGNsRplxNLzlUC-dgbMkB5HdY4uNVhAYKFL7rSLKI2cxckJMkzPOhV9dijkPUiMfG7E316ZHMfzb6pQGn-K_q_9DXYLYW42SM0/w320-h88/overleaf5.png\" title=\"Naming a project in Overleaf\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis95v1JBh55GhkTflZqjpPmi1J8ECZgB7yXamL8LVsEIGNsRplxNLzlUC-dgbMkB5HdY4uNVhAYKFL7rSLKI2cxckJMkzPOhV9dijkPUiMfG7E316ZHMfzb6pQGn-K_q_9DXYLYW42SM0/w320-h88/overleaf5.png\" data-sub-html=\"\u003ch2\u003eNaming a project in Overleaf\u003c/h2\u003e\u003cp\u003eNaming a project in Overleaf\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis95v1JBh55GhkTflZqjpPmi1J8ECZgB7yXamL8LVsEIGNsRplxNLzlUC-dgbMkB5HdY4uNVhAYKFL7rSLKI2cxckJMkzPOhV9dijkPUiMfG7E316ZHMfzb6pQGn-K_q_9DXYLYW42SM0/w320-h88/overleaf5.png' alt=\"Naming a project in Overleaf\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eNaming a project in Overleaf\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis95v1JBh55GhkTflZqjpPmi1J8ECZgB7yXamL8LVsEIGNsRplxNLzlUC-dgbMkB5HdY4uNVhAYKFL7rSLKI2cxckJMkzPOhV9dijkPUiMfG7E316ZHMfzb6pQGn-K_q_9DXYLYW42SM0/s1103/overleaf5.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis95v1JBh55GhkTflZqjpPmi1J8ECZgB7yXamL8LVsEIGNsRplxNLzlUC-dgbMkB5HdY4uNVhAYKFL7rSLKI2cxckJMkzPOhV9dijkPUiMfG7E316ZHMfzb6pQGn-K_q_9DXYLYW42SM0/s1103/overleaf5.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"writing-and-compiling-the-document\" class=\"headerLink\"\u003e\n \u003ca href=\"#writing-and-compiling-the-document\" class=\"header-mark\" aria-label=\"Direct link to section: Writing and Compiling the Document\"\u003e\u003c/a\u003eWriting and Compiling the Document\u003c/h3\u003e\u003ch4 id=\"6-start-writing-and-click-recompile\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-start-writing-and-click-recompile\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Start writing and click Recompile\"\u003e\u003c/a\u003e6. Start writing and click \u003ccode\u003eRecompile\u003c/code\u003e\u003c/h4\u003e\u003cp\u003eThe editor appears on the left side, while the compiled output appears on the right. After writing text or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e commands, click \u003ccode\u003eRecompile\u003c/code\u003e to see the updated result.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjL0aZCN0LZdT2Pod_glB7nZmx-JH6YoVUBqrJEEunxgVLfdsix1ppGLdGjbBxNpRGn4sB0b4wn1fPz_7BvA7vA_-IszdKt0o-O43GUJpE783tITd2pwAd7mn-XhhgSGwC0xCUZ2912gA/w320-h137/overleaf6.png\" title=\"Overleaf editor and compiled output\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjL0aZCN0LZdT2Pod_glB7nZmx-JH6YoVUBqrJEEunxgVLfdsix1ppGLdGjbBxNpRGn4sB0b4wn1fPz_7BvA7vA_-IszdKt0o-O43GUJpE783tITd2pwAd7mn-XhhgSGwC0xCUZ2912gA/w320-h137/overleaf6.png\" data-sub-html=\"\u003ch2\u003eOverleaf editor and compiled output\u003c/h2\u003e\u003cp\u003eOverleaf editor and compiled output\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjL0aZCN0LZdT2Pod_glB7nZmx-JH6YoVUBqrJEEunxgVLfdsix1ppGLdGjbBxNpRGn4sB0b4wn1fPz_7BvA7vA_-IszdKt0o-O43GUJpE783tITd2pwAd7mn-XhhgSGwC0xCUZ2912gA/w320-h137/overleaf6.png' alt=\"Overleaf editor and compiled output\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eOverleaf editor and compiled output\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjL0aZCN0LZdT2Pod_glB7nZmx-JH6YoVUBqrJEEunxgVLfdsix1ppGLdGjbBxNpRGn4sB0b4wn1fPz_7BvA7vA_-IszdKt0o-O43GUJpE783tITd2pwAd7mn-XhhgSGwC0xCUZ2912gA/s1117/overleaf6.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjL0aZCN0LZdT2Pod_glB7nZmx-JH6YoVUBqrJEEunxgVLfdsix1ppGLdGjbBxNpRGn4sB0b4wn1fPz_7BvA7vA_-IszdKt0o-O43GUJpE783tITd2pwAd7mn-XhhgSGwC0xCUZ2912gA/s1117/overleaf6.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"overleaf-menus-and-features\" class=\"headerLink\"\u003e\n \u003ca href=\"#overleaf-menus-and-features\" class=\"header-mark\" aria-label=\"Direct link to section: Overleaf Menus and Features\"\u003e\u003c/a\u003eOverleaf Menus and Features\u003c/h3\u003e\u003ch4 id=\"7-understand-the-main-editor-controls\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-understand-the-main-editor-controls\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Understand the main editor controls\"\u003e\u003c/a\u003e7. Understand the main editor controls\u003c/h4\u003e\u003cp\u003eThe image below highlights the main controls that are commonly used when working in Overleaf.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWgRESAzEgZDXB0sZqYTE1cHBvMCLFYpCGKIj6GVYoaMAj37HCLDjVo9jpkWdSdj1KiMQcYpI1tWEs8M53q5ILMtx49-XSdUjbKeAAjw3YdFVHJ7fw8gGxdWpxExaTVywjKhuUs6fvcxY/w320-h135/overleaf7.png\" title=\"Overleaf editor menu\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWgRESAzEgZDXB0sZqYTE1cHBvMCLFYpCGKIj6GVYoaMAj37HCLDjVo9jpkWdSdj1KiMQcYpI1tWEs8M53q5ILMtx49-XSdUjbKeAAjw3YdFVHJ7fw8gGxdWpxExaTVywjKhuUs6fvcxY/w320-h135/overleaf7.png\" data-sub-html=\"\u003ch2\u003eOverleaf editor menu\u003c/h2\u003e\u003cp\u003eOverleaf editor menu\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWgRESAzEgZDXB0sZqYTE1cHBvMCLFYpCGKIj6GVYoaMAj37HCLDjVo9jpkWdSdj1KiMQcYpI1tWEs8M53q5ILMtx49-XSdUjbKeAAjw3YdFVHJ7fw8gGxdWpxExaTVywjKhuUs6fvcxY/w320-h135/overleaf7.png' alt=\"Overleaf editor menu\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eOverleaf editor menu\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWgRESAzEgZDXB0sZqYTE1cHBvMCLFYpCGKIj6GVYoaMAj37HCLDjVo9jpkWdSdj1KiMQcYpI1tWEs8M53q5ILMtx49-XSdUjbKeAAjw3YdFVHJ7fw8gGxdWpxExaTVywjKhuUs6fvcxY/s1115/overleaf7.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWgRESAzEgZDXB0sZqYTE1cHBvMCLFYpCGKIj6GVYoaMAj37HCLDjVo9jpkWdSdj1KiMQcYpI1tWEs8M53q5ILMtx49-XSdUjbKeAAjw3YdFVHJ7fw8gGxdWpxExaTVywjKhuUs6fvcxY/s1115/overleaf7.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eReference:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eMain menu button.\u003c/li\u003e\n\u003cli\u003eReturn to the project dashboard.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eNew File\u003c/code\u003e to create a new file.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eNew Folder\u003c/code\u003e to create a new folder.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eUpload\u003c/code\u003e to upload files.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eRename\u003c/code\u003e to rename a file or project.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eDelete\u003c/code\u003e to remove a file.\u003c/li\u003e\n\u003cli\u003eThe main \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document name with the \u003ccode\u003e.tex\u003c/code\u003e extension.\u003c/li\u003e\n\u003cli\u003eChange tracking for the document.\u003c/li\u003e\n\u003cli\u003eShare the project by email.\u003c/li\u003e\n\u003cli\u003eSend the document to another service.\u003c/li\u003e\n\u003cli\u003eProject history.\u003c/li\u003e\n\u003cli\u003eCollaboration chat.\u003c/li\u003e\n\u003cli\u003eFull-screen document view.\u003c/li\u003e\n\u003cli\u003eDownload the document.\u003c/li\u003e\n\u003cli\u003eBuild notifications.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eRecompile\u003c/code\u003e button.\u003c/li\u003e\n\u003cli\u003eFit the document to page width.\u003c/li\u003e\n\u003cli\u003eFit the document to page height.\u003c/li\u003e\n\u003cli\u003eZoom in.\u003c/li\u003e\n\u003cli\u003eZoom out.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"practical-tips\" class=\"headerLink\"\u003e\n \u003ca href=\"#practical-tips\" class=\"header-mark\" aria-label=\"Direct link to section: Practical Tips\"\u003e\u003c/a\u003ePractical Tips\u003c/h3\u003e\u003cp\u003eAfter editing the document, get used to pressing \u003ccode\u003eRecompile\u003c/code\u003e often so you can catch errors quickly.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eYellow warnings usually indicate minor issues and can often be ignored temporarily.\u003c/li\u003e\n\u003cli\u003eRed warnings usually indicate fatal errors that must be fixed before the document compiles properly.\u003c/li\u003e\n\u003cli\u003eTo leave Overleaf, open the account menu and choose \u003ccode\u003eLogout\u003c/code\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThat is the basic workflow for writing \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with Overleaf. Once you are comfortable with the editor, you can continue with packages, images, tables, and mathematical notation.\u003c/p\u003e\n", "language": "en" }, { "title" : "Writing LaTeX in TeXLive", "date_published" : "2023-03-01T14:01:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menulis-latex-di-texlive/", "url" : "https://www.aantriono.com/en/menulis-latex-di-texlive/", "summary": " Using TeXLive to Write LaTeX on WindowsTeXLive is one of the most complete LaTeX distributions available for offline use. Its installation size is large, but the advantage is that many packages are already included from the start. Once TeXLive is installed, you can begin writing LaTeX documents with the editors that come with it or with other compatible editors.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"using-texlive-to-write-latex-on-windows\" class=\"headerLink\"\u003e\n \u003ca href=\"#using-texlive-to-write-latex-on-windows\" class=\"header-mark\" aria-label=\"Direct link to section: Using TeXLive to Write LaTeX on Windows\"\u003e\u003c/a\u003eUsing TeXLive to Write LaTeX on Windows\u003c/h2\u003e\u003cp\u003eTeXLive is one of the most complete LaTeX distributions available for offline use. Its installation size is large, but the advantage is that many packages are already included from the start. Once TeXLive is installed, you can begin writing LaTeX documents with the editors that come with it or with other compatible editors.\u003c/p\u003e\n\u003cp\u003eBelow is a short guide for Windows users.\u003c/p\u003e\n\u003ch3 id=\"using-the-texworks-editor\" class=\"headerLink\"\u003e\n \u003ca href=\"#using-the-texworks-editor\" class=\"header-mark\" aria-label=\"Direct link to section: Using the TeXworks Editor\"\u003e\u003c/a\u003eUsing the TeXworks Editor\u003c/h3\u003e\u003cp\u003eFollow these steps to work with TeXworks:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eTurn on your computer.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eClick \u003ccode\u003eStart\u003c/code\u003e \u0026gt; \u003ccode\u003eAll Programs\u003c/code\u003e, find \u003ccode\u003eTeXLive\u003c/code\u003e, and click the arrow icon as shown below.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLGO2e8pF2_w4BAzd-hInzx3C9EfwijPrA96YLoO8f9yDQX2TvEWJbji7uOMvpwHvKCB136hr-ub7rq1ko4JMdxi8xNKldQ3Dp0jK_U9nvObGsfQKop506hUH37svUe6ghbskBEbJI2uM/s682/texlive1.png' alt=\"TeXLive on computer\" \u003e\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eChoose \u003ccode\u003eTeXworks editor\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKv4fECKP2bZyqMUD355H_4GMSWIep1fY4seW2Jk1Dgievk62hWSMYgtXz87ddns7e42GIiXx_DJQrT1cHEKTWloCWcezVbJsYDS63xWVdyxL9R7duUvFzpewNhD-mgcxre2BSY2x-Va8/s665/texlive2.png' alt=\"TeXWorks editor\" \u003e\u003c/p\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eAfter opening it, the interface will look like this.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuRW3Tj13E1xzxYCMuV33CpMBpMdkDglJvuGLF9vmebqIJ2sUo9tef_ftoThd_iuBWKvPg9THkYbSQCthK9WbN3FD9dzKDhFqQKVL4lWz-42EI4zs98-3_01EG5iYstQUFv8GWrrh5zvk/s666/texlive3.png' alt=\"TeXWorks editor interface\" \u003e\u003c/p\u003e\n\u003cp\u003eThe white area is where you type your LaTeX source, and the green button is used to compile the document.\u003c/p\u003e\n\u003ch3 id=\"using-the-texstudio-editor\" class=\"headerLink\"\u003e\n \u003ca href=\"#using-the-texstudio-editor\" class=\"header-mark\" aria-label=\"Direct link to section: Using the TeXstudio Editor\"\u003e\u003c/a\u003eUsing the TeXstudio Editor\u003c/h3\u003e\u003cp\u003eIf you prefer another editor, such as TeXstudio, it also works well with TeXLive.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eTurn on your computer.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eClick \u003ccode\u003eStart\u003c/code\u003e \u0026gt; \u003ccode\u003eAll Programs\u003c/code\u003e, then search for \u003ccode\u003eTeXstudio\u003c/code\u003e.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikjgp1zaWph2Ai5QC447bK5GFmiV7rbfEVzzh8THOBI_YKyznt-1EZ3qmyuHfDAbgxnuY7PA_HWLuntmeeYG7aB9Gc5SzrDVTeUO4GX17ueV72xRDDF6zUzPGargTuoExcD4dX1vCVHKE/s645/texlive5.png' alt=\"TeXstudio editor\" \u003e\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eAfter opening TeXstudio, the interface will look like this.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuK1ZnQI-9f-baOF8dkqWnkLi8J2SnyYuF4PJ0qTndtZ2EeObglS0WT9a_U18Rm2q26u2u041V6bi-fSBUpCgiSQIOmnoUyZrqnqpdIFLCl8QCCbH1u8MK4QwegVh26xq_I8d9mE5XWUQ/s1209/texlive4.png' alt=\"TeXstudio with TeXLive\" \u003e\u003c/p\u003e\n\u003cp\u003eTeXstudio exposes more features directly in the interface. To create a new document, you can:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003epress \u003ccode\u003eCtrl+N\u003c/code\u003e, or\u003c/li\u003e\n\u003cli\u003eclick \u003ccode\u003eFile \u0026gt; New\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThen write your \u003ccode\u003e.tex\u003c/code\u003e file as usual. When you are done:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eclick \u003ccode\u003eSave\u003c/code\u003e or \u003ccode\u003eSave As\u003c/code\u003e to store the file\u003c/li\u003e\n\u003cli\u003eclick the green triangle button to compile the document\u003c/li\u003e\n\u003cli\u003ewait until the generated output appears\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat is a short guide to using TeXLive on Windows with TeXworks and TeXstudio. After installation, the next practical step is simply choosing the editor that best fits your workflow.\u003c/p\u003e\n", "language": "en" }, { "title" : "Relationship of Angles on Two Parallel Lines Intersected by Another Line", "date_published" : "2023-03-01T14:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/hubungan-sudut-sudut-pada-dua-garis/", "url" : "https://www.aantriono.com/en/hubungan-sudut-sudut-pada-dua-garis/", "summary": " Drawing and Animating Angle Relationships on Two Parallel LinesThis article explains how to draw and animate angle relationships on two parallel lines intersected by another line using GeoGebra. This topic is commonly studied in grade VII mathematics.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-and-animating-angle-relationships-on-two-parallel-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-and-animating-angle-relationships-on-two-parallel-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing and Animating Angle Relationships on Two Parallel Lines\"\u003e\u003c/a\u003eDrawing and Animating Angle Relationships on Two Parallel Lines\u003c/h2\u003e\u003cp\u003eThis article explains how to draw and animate angle relationships on two parallel lines intersected by another line using GeoGebra. This topic is commonly studied in grade VII mathematics.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_Fzxx4sctWM8hTx3hMi-QbVywafBPoU82HGCM0OWJI3UlVX5ZUjqV30zB0Jt3r9snHzVMKg296zfJUjPAe9wM6AZoBs_Fo6paLX-m_PIDRbiiPt-iBzgG0mZ7M_pF5UtudUE13LFGePg/s519/sudut.png' alt=\"Angle relationships on two parallel lines\" \u003e\u003c/p\u003e\n\u003ch3 id=\"drawing-angles-on-two-parallel-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-angles-on-two-parallel-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Angles on Two Parallel Lines\"\u003e\u003c/a\u003eDrawing Angles on Two Parallel Lines\u003c/h3\u003e\u003cp\u003eThe following steps show how to draw two parallel lines, a transversal, and the related angles.\u003c/p\u003e\n\u003ch4 id=\"steps-1-3-creating-parallel-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-1-3-creating-parallel-lines\" class=\"header-mark\" aria-label=\"Direct link to section: Steps 1-3: Creating Parallel Lines\"\u003e\u003c/a\u003eSteps 1-3: Creating Parallel Lines\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eOpen GeoGebra.\u003c/li\u003e\n\u003cli\u003eDraw a line through points A and B. Click \u003cem\u003eLine\u003c/em\u003e, then create points such as A \u003ccode\u003e(-6, 5)\u003c/code\u003e and B \u003ccode\u003e(2, 5)\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eClick \u003cem\u003eParallel Line\u003c/em\u003e, then click line \u003ccode\u003eAB\u003c/code\u003e. Place the new parallel line through point C \u003ccode\u003e(-3, 2)\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguBTySSNtzs2FA-J2x0kxAqz3piRiZdAk49SsDgj29ehwd929m5nqdMwopki-WRkgD93n6v0AN7CsvWd2Xpe7WEBx0TViY1oinquakBKThSzrO2EmRyBopzdLtivlghsxL6g9DtghGfa8/s1266/sudut1.png' alt=\"Creating parallel lines\" \u003e\u003c/p\u003e\n\u003ch4 id=\"steps-4-5-creating-the-transversal-and-points\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-4-5-creating-the-transversal-and-points\" class=\"header-mark\" aria-label=\"Direct link to section: Steps 4-5: Creating the Transversal and Points\"\u003e\u003c/a\u003eSteps 4-5: Creating the Transversal and Points\u003c/h4\u003e\u003col start=\"4\"\u003e\n\u003cli\u003eDraw a line passing through points A and C so that it intersects both parallel lines.\u003c/li\u003e\n\u003cli\u003eAdd points on the lines, for example D, E, F, G, H, and I. These points are used to measure angles and observe their relationships.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZKV3Usj2xI6tJZoXlWjbppsC4P5hV7cIU-jkqJgK0k7_cWt_n83eKOpx-8-i14HL7aHSn2SG2DaolrDbEd8oO8Co6wbHTrfycOY-yHhsnP4xWR3Kog33lZ8yi7L8AFRksIjBH_s2f9qU/s1271/sudut2.png' alt=\"Two parallel lines cut by another line\" \u003e\u003c/p\u003e\n\u003ch4 id=\"steps-6-7-creating-and-tidying-the-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-6-7-creating-and-tidying-the-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Steps 6-7: Creating and Tidying the Angles\"\u003e\u003c/a\u003eSteps 6-7: Creating and Tidying the Angles\u003c/h4\u003e\u003col start=\"6\"\u003e\n\u003cli\u003eClick \u003cem\u003eAngle\u003c/em\u003e and create the angles \u003ccode\u003eDAE\u003c/code\u003e, \u003ccode\u003eEAF\u003c/code\u003e, \u003ccode\u003eFAC\u003c/code\u003e, \u003ccode\u003eCAD\u003c/code\u003e, \u003ccode\u003eHCA\u003c/code\u003e, \u003ccode\u003eACG\u003c/code\u003e, \u003ccode\u003eGCI\u003c/code\u003e, and \u003ccode\u003eICH\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eTidy the figure by giving equal angles the same color.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQumYNxjvvqSt4XkuWOq68P485PrrbjAjjhsEcofppzSChTA2J1ktp6tyF59AE52gaph3CtQdEOWYRwwQHLu6K2wLpaz3L6ygIprgt5X-rUu2SeRu3sn30vjGGUNhraWNmqb8H1BTL6zs/s485/sudut3.png' alt=\"Angles on two parallel lines\" \u003e\u003c/p\u003e\n\u003ch3 id=\"creating-animations-of-angle-relationships\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-animations-of-angle-relationships\" class=\"header-mark\" aria-label=\"Direct link to section: Creating Animations of Angle Relationships\"\u003e\u003c/a\u003eCreating Animations of Angle Relationships\u003c/h3\u003e\u003cp\u003eAfter the base figure is ready, create animations for several angle pairs.\u003c/p\u003e\n\u003ch4 id=\"animation-of-corresponding-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#animation-of-corresponding-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Animation of Corresponding Angles\"\u003e\u003c/a\u003eAnimation of Corresponding Angles\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eClick \u003cem\u003eSlider\u003c/em\u003e, then choose \u003cem\u003eCheck Box\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eClick on the coordinate plane to place the \u003cem\u003eCheck Box\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eFill the \u003cem\u003eCaption\u003c/em\u003e field with \u003ccode\u003eCorresponding angles pair 1\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eClick the small arrow below \u003cem\u003eCaption\u003c/em\u003e, then choose \u003cem\u003eAngle between D,A,E\u003c/em\u003e and \u003cem\u003eAngle between H,C,A\u003c/em\u003e. Click \u003cem\u003eOK\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eWhen the check mark is clicked, the corresponding angles disappear. Click the \u003cem\u003eCheck Box\u003c/em\u003e again to show them.\u003c/li\u003e\n\u003cli\u003eRepeat the same steps for the other corresponding pairs: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle EAF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle ACG\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eCG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle FAC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle GCI\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eGC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07847em;\"\u003eI\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle CAD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eI\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle ICH\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07847em;\"\u003eI\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifY0tSPpGqOr5QfX0ierjgfNMGDLk2heurjWF3K83sfgLHaCe2TTWgU7XJIHSSdw_HAKeGug9toC4Luxxt7-3FEld2RQ1JaGCHHIpcXCQUWT7Q8ZQZZgSfuanXUAoMQN1mwsqbUcXszrM/s674/sudut4.png' alt=\"Setting corresponding angle pairs\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghnscJn28HxgwPe-3pGwFDr32f7-Fu67hSfevJF95LzvoppoLjnXRAU-9p5gO-GwrMyQnkRLE_5QOBg2GnVIzJ0AhJDZXOzX66rUkF1f9ULukZSO94F2c6moB8s1xdr3mgruS8tG47xOY/s435/sudut5.png' alt=\"Selecting corresponding angles\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg74LxCCnloP8XF-R7S4T16Huyv_u6PB8qvt2O_8WNed2Ra0mbLCs0rVDCNzaBw8bHNeDWTdBUPmyPkBukXUTbsQoq8bmrrtFPm9q0KbUH3yuI1uv3T4_dan8A4RHBIC0YnRQ5MvZSiM80/s650/sudut6.png' alt=\"Result of corresponding angles\" \u003e\u003c/p\u003e\n\u003cp\u003eTry the animation below.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/f9ygmw9a/width/600/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; max-width: 100%;\" title=\"Pasangan sudut sehadap\" width=\"600px\"\u003e \u003c/iframe\u003e\n\u003ch4 id=\"animation-of-alternate-interior-and-alternate-exterior-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#animation-of-alternate-interior-and-alternate-exterior-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Animation of Alternate Interior and Alternate Exterior Angles\"\u003e\u003c/a\u003eAnimation of Alternate Interior and Alternate Exterior Angles\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eClick \u003cem\u003eSlider\u003c/em\u003e, then choose \u003cem\u003eCheck Box\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eClick on the coordinate plane to place the \u003cem\u003eCheck Box\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eFill the \u003cem\u003eCaption\u003c/em\u003e field with \u003ccode\u003eAlternate interior angles pair 1\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eSelect \u003cem\u003eAngle between F,A,C\u003c/em\u003e and \u003cem\u003eAngle between H,C,A\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eAnother alternate interior pair is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle DAC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle ACG\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eCG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eFill the \u003cem\u003eCaption\u003c/em\u003e field with \u003ccode\u003eAlternate exterior angles pair 1\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eSelect \u003cem\u003eAngle between D,A,E\u003c/em\u003e and \u003cem\u003eAngle between G,C,I\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eAnother alternate exterior pair is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle EAF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle HCI\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07847em;\"\u003eI\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eTry the animation below.\u003c/p\u003e\n\u003ciframe height=\"465px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/rhuddj5n/width/620/height/465/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; max-width: 100%;\" title=\"Pasangan sudut berseberangan\" width=\"620px\"\u003e \u003c/iframe\u003e\n\u003ch4 id=\"animation-of-same-side-interior-and-same-side-exterior-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#animation-of-same-side-interior-and-same-side-exterior-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Animation of Same-Side Interior and Same-Side Exterior Angles\"\u003e\u003c/a\u003eAnimation of Same-Side Interior and Same-Side Exterior Angles\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eClick \u003cem\u003eSlider\u003c/em\u003e, then choose \u003cem\u003eCheck Box\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eClick on the coordinate plane to place the \u003cem\u003eCheck Box\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eFill the \u003cem\u003eCaption\u003c/em\u003e field with \u003ccode\u003eSame-side interior angles pair 1\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eSelect \u003cem\u003eAngle between D,A,C\u003c/em\u003e and \u003cem\u003eAngle between H,C,A\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eAnother same-side interior pair is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle FAC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle ACG\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eCG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eFill the \u003cem\u003eCaption\u003c/em\u003e field with \u003ccode\u003eSame-side exterior angles pair 1\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eSelect \u003cem\u003eAngle between D,A,E\u003c/em\u003e and \u003cem\u003eAngle between I,C,H\u003c/em\u003e.\u003c/li\u003e\n\u003cli\u003eAnother same-side exterior pair is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle EAF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\angle GCI\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eGC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07847em;\"\u003eI\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eTry the animation below.\u003c/p\u003e\n\u003ciframe height=\"450px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/gtqpj3bc/width/600/height/450/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; max-width: 100%;\" title=\"Sudut dalam sepihak\" width=\"600px\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat concludes this guide to drawing and animating angle relationships on two parallel lines intersected by another line in GeoGebra. If you have suggestions or corrections, please leave them in the comments or contact me \u003ca href=\"/p/contact.html\" rel=\"\"\u003ehere\u003c/a\u003e. More GeoGebra animation examples can also be found on \u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ethis website\u003c/a\u003e.\u003c/p\u003e\n", "language": "en" }, { "title" : "Shadow of Two Circles", "date_published" : "2023-03-01T14:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bayangan-dua-lingkaran/", "url" : "https://www.aantriono.com/en/bayangan-dua-lingkaran/", "summary": " Creating a Two-Circle Shadow Animation in GeoGebraThis tutorial shows how to create a moving shadow-like pattern from two circles in GeoGebra. The effect comes from animating a point on one circle, building another circle from that moving point, and letting its trace produce a colorful pattern.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"creating-a-two-circle-shadow-animation-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-a-two-circle-shadow-animation-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Creating a Two-Circle Shadow Animation in GeoGebra\"\u003e\u003c/a\u003eCreating a Two-Circle Shadow Animation in GeoGebra\u003c/h2\u003e\u003cp\u003eThis tutorial shows how to create a moving shadow-like pattern from two circles in GeoGebra. The effect comes from animating a point on one circle, building another circle from that moving point, and letting its trace produce a colorful pattern.\u003c/p\u003e\n\u003cp\u003eHere is the visual result.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4oNsoPK699ToNWkqbSNsNXddpvkMd6uVuJAJ-35GfaS6s89aClWH8av4192Mu7ytDOtnpG7-_u5vaB4chhchnPUxZLjEE6yajQCVlb143vgP0n07UK42hdcWqA0Wv-C6RZUc26n3IYpI/w320-h262/bayangan.png' alt=\"two-circle shadow animation\" \u003e\u003c/p\u003e\n\u003ch3 id=\"steps-1-3-create-the-first-circle-and-point-c\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-1-3-create-the-first-circle-and-point-c\" class=\"header-mark\" aria-label=\"Direct link to section: Steps 1-3: Create the First Circle and Point C\"\u003e\u003c/a\u003eSteps 1-3: Create the First Circle and Point C\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen GeoGebra.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSelect \u003cem\u003eCircle with Center through Point\u003c/em\u003e, then create a circle with center \u003ccode\u003eA\u003c/code\u003e and point \u003ccode\u003eB\u003c/code\u003e.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS8JfWHZ0ahaC4_VgOhkdahjg3LLNz46vbJCZCXDFmIIb_IItHiyLkhvu53ou9dIFq19jTZTXnMtOHYtywQsLX1BDsLDVECv8jCRRf1syPBulK7b5Hq7OtOLIUcW0BT5BqHoJIDt6g828/w320-h173/bayangan1.png' alt=\"circle\" \u003e\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eSelect \u003cem\u003eNew Point\u003c/em\u003e, then place point \u003ccode\u003eC\u003c/code\u003e on the circumference.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2XTGqwI_Yu8TAR_5TMBuS3yT07JOZdrRm3NsbqOo5lM_tu8przN87FbfmNiiUc-8ip8VKC8_4m-QDJCgn3iHsuUK_MMSrcc_3zYFPkywqQGuPDLZ8rvKudT0Nc86O9umWGmqOlQuWKrE/w320-h190/bayangan11.png' alt=\"point C on the circle\" \u003e\u003c/p\u003e\n\u003ch3 id=\"steps-4-5-create-the-second-circle-and-turn-on-animation\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-4-5-create-the-second-circle-and-turn-on-animation\" class=\"header-mark\" aria-label=\"Direct link to section: Steps 4-5: Create the Second Circle and Turn On Animation\"\u003e\u003c/a\u003eSteps 4-5: Create the Second Circle and Turn On Animation\u003c/h3\u003e\u003col start=\"4\"\u003e\n\u003cli\u003eSelect \u003cem\u003eCircle with Center through Point\u003c/em\u003e again, then click \u003ccode\u003eC\u003c/code\u003e as the center and \u003ccode\u003eB\u003c/code\u003e as the point on the circle.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXrma5-mOCj0l0GTcqkB5cJ8sC0BKCqV5u02EMTvTN_Vxl0x4H-3b8W3lDpgN6xRImFa1f6TXMxd7MLn4H1XqmUy2jYiFxkotbJJM_ndPH37NQKiasxKSAcLOcobQI5ccfXDC30dc4g90/w320-h199/bayangan12.png' alt=\"two circles\" \u003e\u003c/p\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eRight-click point \u003ccode\u003eC\u003c/code\u003e, then choose \u003cem\u003eAnimation On\u003c/em\u003e. The second circle will now move.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgri4qq-kn9ZPNou35oGkXfY_JpaJ6EaHttivOsfH9I3pJ_AUu55ryUw0QBK43h66_NfosN9oifjQGtjE4lHmUt2peMufoCSKt8A3Irk5xqlX_jPVR2PIbEap6K8NClVq61KzgBMEIgupY/w320-h468/bayangan3.png' alt=\"Animation On for point C\" \u003e\u003c/p\u003e\n\u003ch3 id=\"steps-6-7-set-trace-and-dynamic-color\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-6-7-set-trace-and-dynamic-color\" class=\"header-mark\" aria-label=\"Direct link to section: Steps 6-7: Set Trace and Dynamic Color\"\u003e\u003c/a\u003eSteps 6-7: Set Trace and Dynamic Color\u003c/h3\u003e\u003col start=\"6\"\u003e\n\u003cli\u003eChoose the \u003cem\u003eMove\u003c/em\u003e tool, right-click the circle centered at \u003ccode\u003eC\u003c/code\u003e, open \u003cem\u003eSettings\u003c/em\u003e, and enable \u003cem\u003eShow Trace\u003c/em\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxCL8ovaEnXmU6DJq9VC5n5x6wQiKVYphjbzYce8-aOUWEv43n8Rs8kxPM2t222Al-ihbvrNpLBOm15LwEd4RiTKFPkcIc7jbYQ9rowv_ifVz9AY-FkwOjPUr93rC7Fea9Ld3ZbF57Kzc/w320-h130/bayangan5.png' alt=\"show trace\" \u003e\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eIn the \u003cem\u003eAdvanced\u003c/em\u003e tab, fill the dynamic color fields with:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003eRed = x(C)\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eGreen = y(C)\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003eBlue = 4x(C)\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzXq9RtgbmVuAQtkZGwlDQOzBD16beihF6W1qzyba6gLJFmvofS8swZ8GBnfS62GvpPTnYdXYfwfMUGGf4kV0QwScmkg6TsAi4N9CHzd8WaEMlRbR47DeEzijqf_hlaHZ4sYXpKNZp5nI/w320-h112/bayangan6.png' alt=\"advanced RGB settings\" \u003e\u003c/p\u003e\n\u003ch3 id=\"steps-8-12-finish-and-explore-the-pattern\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-8-12-finish-and-explore-the-pattern\" class=\"header-mark\" aria-label=\"Direct link to section: Steps 8-12: Finish and Explore the Pattern\"\u003e\u003c/a\u003eSteps 8-12: Finish and Explore the Pattern\u003c/h3\u003e\u003col start=\"8\"\u003e\n\u003cli\u003e\n\u003cp\u003eReturn to the basic settings and confirm that \u003cem\u003eShow Trace\u003c/em\u003e is still enabled.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eTurn \u003cem\u003eAnimation On\u003c/em\u003e for point \u003ccode\u003eC\u003c/code\u003e again if needed.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfvwZtqu6w1AHDLKRBPYCF2CpwMthldnUQ9KIZL8Rb9S4glOiVktd821sexK69HwMggGIh-ndyVCtk9I631La1K-Xs3mo-8fsv6mz8l9DwkVritwcKhnTt1Mn4PtakClIBG79btsqrvtE/w320-h111/bayangan7.png' alt=\"animation on for the circle\" \u003e\u003c/p\u003e\n\u003col start=\"10\"\u003e\n\u003cli\u003e\n\u003cp\u003eOpen \u003cem\u003eView\u003c/em\u003e and choose \u003cem\u003eRefresh Views\u003c/em\u003e to clear and reset the display.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eLet the animation run again to build a new trace pattern.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eRepeat \u003cem\u003eRefresh Views\u003c/em\u003e whenever you want to generate another variation.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhldorVjwYb9jbyza5wr_t23_B21Kn83vPbF0P8YD7fhV0zJrEvlkpsyixj_7qx2xVyBYI3AYCTx5GAq0LmgQWC-99tZQFhErWL60u4CBYCmfijg50XwLrYwmKrOs4adSDsYCnn4rUB9N8/w320-h169/bayangan9.png' alt=\"refresh views\" \u003e\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1Gw54nCJox3ozW9hewlmZLevLHtdimgNIxF0wGeVxev6YRVWUj2zFkFFmqEFUITkwI6w0w2t11CClP32TMWZLI8RrOinNqgxC_sRdqFVTiliOEkvctNhNwewI9sEVz0NLgHwtDvYBvxI/w320-h169/bayangan10.png' alt=\"two-circle shadow pattern\" \u003e\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-7SVjlj7l2twKWUbMBNJMwhUiYJH5abMXdMv3feQmA7DutMr07KbEF1whANs8Mxyb5l9IAxg35psaWQ3N6XyXjgeBUgLlIFeeBVB1JEEp2WpOFGMwpfOKcQ7JrT9_VtA-3VHxz6FgTJo/w320-h168/bayangan8.png' alt=\"animated two-circle shadow\" \u003e\u003c/p\u003e\n\u003ch3 id=\"final-interactive-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#final-interactive-result\" class=\"header-mark\" aria-label=\"Direct link to section: Final Interactive Result\"\u003e\u003c/a\u003eFinal Interactive Result\u003c/h3\u003e\u003cp\u003eYou can try the interactive version below. Press the \u003cem\u003ePlay\u003c/em\u003e button in the lower-left corner, then drag the points or circles to see different shadow variations.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/qnfn6veh/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; max-width: 100%;\" title=\"Bayangan Dua Lingkaran\" width=\"650px\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat is the basic workflow for creating a two-circle shadow animation in GeoGebra. By combining point animation, trace, and dynamic color, you can produce a simple but visually rich motion pattern.\u003c/p\u003e\n", "language": "en" }, { "title" : "LaTeX Script Structure", "date_published" : "2023-03-01T13:59:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/struktur-naskah-latex/", "url" : "https://www.aantriono.com/en/struktur-naskah-latex/", "summary": " LaTeX Script StructureAfter installing LaTeX\\LaTeXLATE​X on your computer and getting familiar with document classes and packages, the next step is to start writing documents with a clear structure.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"latex-script-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-script-structure\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Script Structure\"\u003e\u003c/a\u003eLaTeX Script Structure\u003c/h2\u003e\u003cp\u003eAfter installing \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e on your computer and getting familiar with document classes and \u003cem\u003epackages\u003c/em\u003e, the next step is to start writing documents with a clear structure.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioBg0aro4VjrAhDzrQBRvKpaoQWkocAY26s8ym0FctRwfhx9yKDkLnGmqtITB0BYKAUSUZ4ielxIEyo5x-w6s2T8S6lT-nx5HMC75ny0RKcBKbL53qNjZsESkvdo6y_CKsluirAXIkkWs/s527/struktur+naskah.png' alt=\"LaTeX script structure\" \u003e\u003c/p\u003e\n\u003cp\u003eIn general, a \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document structure may include the document title, paragraphs, chapters, sections, comments, table of contents, figures, tables, and other supporting elements.\u003c/p\u003e\n\u003ch3 id=\"document-title\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-title\" class=\"header-mark\" aria-label=\"Direct link to section: Document Title\"\u003e\u003c/a\u003eDocument Title\u003c/h3\u003e\u003cp\u003eThe title, author name, and date can be written like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eDocument Title\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAuthor Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThen display them in the document body with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\maketitle\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you do not want to use the current date, replace \u003ccode\u003e\\today\u003c/code\u003e with any date you want.\u003c/p\u003e\n\u003ch3 id=\"paragraphs\" class=\"headerLink\"\u003e\n \u003ca href=\"#paragraphs\" class=\"header-mark\" aria-label=\"Direct link to section: Paragraphs\"\u003e\u003c/a\u003eParagraphs\u003c/h3\u003e\u003cp\u003eIn a \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document, paragraphs are separated by one or more blank lines, or by using:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\par\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you want to start a new line without starting a new paragraph, use:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eor\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newline\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTo begin a new page, the common commands are:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\clearpage\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\cleardoublepage\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\clearemptydoublepage\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eCommands beginning with \u003ccode\u003e\\clear\u003c/code\u003e will place pending \u003cem\u003efloats\u003c/em\u003e such as figures or tables before creating the new page. \u003ccode\u003e\\cleardoublepage\u003c/code\u003e is often used in book-style documents so a new major section starts on the right-hand page.\u003c/p\u003e\n\u003ch3 id=\"chapters-subchapters-and-sections\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapters-subchapters-and-sections\" class=\"header-mark\" aria-label=\"Direct link to section: Chapters, Subchapters, and Sections\"\u003e\u003c/a\u003eChapters, Subchapters, and Sections\u003c/h3\u003e\u003cp\u003eFor the \u003ccode\u003earticle\u003c/code\u003e document class, the common structure commands are:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSection Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\subsection\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSubsection Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\subsubsection\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSubsubsection Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\paragraph\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eParagraph Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\subparagraph\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSubparagraph Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eFor the \u003ccode\u003ereport\u003c/code\u003e and \u003ccode\u003ebook\u003c/code\u003e document classes, there are additional commands:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\part\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ePart Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\chapter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eChapter Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThese commands automatically generate numbering. If you want an unnumbered heading, use the starred form, for example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section*\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\chapter*\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ePreface\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the \u003ccode\u003ebook\u003c/code\u003e class, large document divisions are usually marked with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\frontmatter\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\mainmatter\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\backmatter\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTheir roles are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e\\frontmatter\u003c/code\u003e for preliminary pages such as the title page, preface, and table of contents,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\mainmatter\u003c/code\u003e for the main content,\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e\\backmatter\u003c/code\u003e for the final section such as bibliography, index, or glossary.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"example-of-an-article-document-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-of-an-article-document-structure\" class=\"header-mark\" aria-label=\"Direct link to section: Example of an article Document Structure\"\u003e\u003c/a\u003eExample of an \u003ccode\u003earticle\u003c/code\u003e Document Structure\u003c/h4\u003e\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eLearning LaTeX\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAan Triono\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\maketitle\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eIntroduction text.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eDiscussion\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\subsection\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eFirst Topic\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eFirst topic text.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\subsection\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eSecond Topic\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eSecond topic text.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eClosing\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eClosing text.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch4 id=\"example-of-a-book-document-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-of-a-book-document-structure\" class=\"header-mark\" aria-label=\"Direct link to section: Example of a book Document Structure\"\u003e\u003c/a\u003eExample of a \u003ccode\u003ebook\u003c/code\u003e Document Structure\u003c/h4\u003e\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebook\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eBook Title\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAuthor Name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\frontmatter\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\maketitle\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tableofcontents\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\mainmatter\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\chapter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eFirst chapter content.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\chapter\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eDiscussion\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eTopic One\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003eTopic one content.\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\backmatter\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\chapter*\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eBibliography\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"comments\" class=\"headerLink\"\u003e\n \u003ca href=\"#comments\" class=\"header-mark\" aria-label=\"Direct link to section: Comments\"\u003e\u003c/a\u003eComments\u003c/h3\u003e\u003cp\u003eIn \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 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style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, comments start with the percent symbol \u003ccode\u003e%\u003c/code\u003e and continue to the end of the line.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n 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26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e% This is a comment and will not be printed\n\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"c\"\u003e\u003c/span\u003e\u003cspan class=\"k\"\u003e\\section\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eIntroduction\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eComments are useful for notes or for temporarily disabling code.\u003c/p\u003e\n\u003ch3 id=\"table-of-contents-list-of-figures-and-list-of-tables\" class=\"headerLink\"\u003e\n \u003ca href=\"#table-of-contents-list-of-figures-and-list-of-tables\" class=\"header-mark\" aria-label=\"Direct link to section: Table of Contents, List of Figures, and List of Tables\"\u003e\u003c/a\u003eTable of Contents, List of Figures, and List of Tables\u003c/h3\u003e\u003cp\u003eIf your document uses structural commands such as \u003ccode\u003e\\part\u003c/code\u003e, \u003ccode\u003e\\chapter\u003c/code\u003e, and \u003ccode\u003e\\section\u003c/code\u003e, the table of contents can be generated automatically with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n 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\n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tableofcontents\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf the document contains many figures and tables, you can also generate their lists with:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\listoffigures\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\listoftables\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat is a short overview of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e document structure. Once you understand the basic layout, writing a neat and consistent document becomes much easier.\u003c/p\u003e\n", "language": "en" }, { "title" : "Love in GeoGebra", "date_published" : "2023-03-01T13:59:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/love-in-geogebra/", "url" : "https://www.aantriono.com/en/love-in-geogebra/", "summary": " Creating a LOVE Graph in GeoGebraThis tutorial shows how to create a graph that looks like the word LOVE in GeoGebra. Each letter is formed from a different equation, then the final result is refined with color, line thickness, and label settings.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"creating-a-love-graph-in-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-a-love-graph-in-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Creating a LOVE Graph in GeoGebra\"\u003e\u003c/a\u003eCreating a LOVE Graph in GeoGebra\u003c/h2\u003e\u003cp\u003eThis tutorial shows how to create a graph that looks like the word \u003cstrong\u003eLOVE\u003c/strong\u003e in GeoGebra. Each letter is formed from a different equation, then the final result is refined with color, line thickness, and label settings.\u003c/p\u003e\n\u003cp\u003eHere is an example of the final result.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaBX4gfvEbG4M8qIHyi4opOBDCqVzFAS374s2dZpJQc55saewVKPEv0n9KAqJDpxPYlbnotIP0tJQn-6mziftvoAIcIbGodiUMnJsdxLVgutCiC13EgjsF0Htl-3G108IYMynyaTyZSsI/s640/love.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"483\" data-original-width=\"640\" height=\"242\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaBX4gfvEbG4M8qIHyi4opOBDCqVzFAS374s2dZpJQc55saewVKPEv0n9KAqJDpxPYlbnotIP0tJQn-6mziftvoAIcIbGodiUMnJsdxLVgutCiC13EgjsF0Htl-3G108IYMynyaTyZSsI/w320-h242/love.png\" title=\"love graph in geogebra\" width=\"320\" style=\"max-width: 100%; height: auto;\" alt=\"Example of a LOVE graph in GeoGebra\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"steps-to-create-the-love-graph\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-create-the-love-graph\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Create the LOVE Graph\"\u003e\u003c/a\u003eSteps to Create the LOVE Graph\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen GeoGebra.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eTo form the letter \u003cstrong\u003eL\u003c/strong\u003e, enter the following equation in the \u003cem\u003eInput\u003c/em\u003e field:\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ey = 1/x\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5VHovaYXDYWDmxres8naJDYfKpvx8tdDO4qpqscFTmn3_pw8aXjp9tNYWhJan4XUF7nOIPuwebMtF3SOW5R438ZWLishq_M-o1p7GkzsHYP2-x5mgQ1dDEo_kpXC1fIqzPSBPXyCg5Wg/s682/love1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"graph of y=1/x in geogebra\" border=\"0\" data-original-height=\"478\" data-original-width=\"682\" height=\"224\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5VHovaYXDYWDmxres8naJDYfKpvx8tdDO4qpqscFTmn3_pw8aXjp9tNYWhJan4XUF7nOIPuwebMtF3SOW5R438ZWLishq_M-o1p7GkzsHYP2-x5mgQ1dDEo_kpXC1fIqzPSBPXyCg5Wg/w320-h224/love1.png\" title=\"graph of y=1/x\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eTo form the letter \u003cstrong\u003eO\u003c/strong\u003e, enter:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ex^2 + y^2 = 2\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eYou can change the constant \u003ccode\u003e2\u003c/code\u003e if you want a different circle size.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgO7jjeL_Bh427f81pwlBoHDjwRvrvmSv57CdJ_j4qvRdMM16f_9cFjFlyPTiAxUqVe4RpyFy3Sb-yMDWVBtD4mcnNrVAf9Y0E60BpOlCfAnQ0Dwyo7MGPlI_kawEjvd184rhYOEcH1yGQ/s676/love2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"graph of x^2+y^2=2 in geogebra\" border=\"0\" data-original-height=\"476\" data-original-width=\"676\" height=\"225\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgO7jjeL_Bh427f81pwlBoHDjwRvrvmSv57CdJ_j4qvRdMM16f_9cFjFlyPTiAxUqVe4RpyFy3Sb-yMDWVBtD4mcnNrVAf9Y0E60BpOlCfAnQ0Dwyo7MGPlI_kawEjvd184rhYOEcH1yGQ/w320-h225/love2.png\" title=\"graph of x^2+y^2=2\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eTo form the letter \u003cstrong\u003eV\u003c/strong\u003e, enter:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ey = |-2x|\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8yi580QcVstm5mir_NuFdCFupIww4bm4xqF4a9mG6x8EC8J1y7V7A9cFwBbT_KLN1Uim4pbTkWr8QpEAEciN8J_QoBiAqaae-Ghe3MxdxVMXfmvN9EY4h0O9coWcRL3cjokP29bgVPao/s765/love3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"graph of y=|-2x| in geogebra\" border=\"0\" data-original-height=\"480\" data-original-width=\"765\" height=\"201\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8yi580QcVstm5mir_NuFdCFupIww4bm4xqF4a9mG6x8EC8J1y7V7A9cFwBbT_KLN1Uim4pbTkWr8QpEAEciN8J_QoBiAqaae-Ghe3MxdxVMXfmvN9EY4h0O9coWcRL3cjokP29bgVPao/w320-h201/love3.png\" title=\"graph of y=|-2x|\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eTo form the letter \u003cstrong\u003eE\u003c/strong\u003e, enter:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003etext\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ex = -3 sin(y)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNleTcKW09zWsbIeMZnhrAWVSKoCjuv9_gUor26ycSWz6qpHrYm4gJSjg04nanny2w7gsxQy6B2QjNYexekxoTEIwCfo7c0x8O8LVxkWfZ_oawjPl29pNtAIpdt8UYzewQIkfZDGk2NhQ/s719/love4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"graph of x=-3sin(y) in geogebra\" border=\"0\" data-original-height=\"494\" data-original-width=\"719\" height=\"220\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNleTcKW09zWsbIeMZnhrAWVSKoCjuv9_gUor26ycSWz6qpHrYm4gJSjg04nanny2w7gsxQy6B2QjNYexekxoTEIwCfo7c0x8O8LVxkWfZ_oawjPl29pNtAIpdt8UYzewQIkfZDGk2NhQ/w320-h220/love4.png\" title=\"graph of x=-3sin(y)\" width=\"320\" style=\"max-width: 100%; height: auto;\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eAs a final step, change the color of each graph, adjust the line thickness, remove labels if necessary, and refine the layout so each letter is easier to read.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"love-graph-result\" class=\"headerLink\"\u003e\n \u003ca href=\"#love-graph-result\" class=\"header-mark\" aria-label=\"Direct link to section: LOVE Graph Result\"\u003e\u003c/a\u003eLOVE Graph Result\u003c/h3\u003e\u003cp\u003eYou can see an interactive example below.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/vgkgtqxq/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px; max-width: 100%;\" title=\"Love\" width=\"650px\"\u003e \u003c/iframe\u003e\n\u003cp\u003eDo you think the graph already looks like the word \u0026ldquo;LOVE\u0026rdquo;?\u003c/p\u003e\n\u003ch3 id=\"another-love-variation\" class=\"headerLink\"\u003e\n \u003ca href=\"#another-love-variation\" class=\"header-mark\" aria-label=\"Direct link to section: Another LOVE Variation\"\u003e\u003c/a\u003eAnother LOVE Variation\u003c/h3\u003e\u003cp\u003eHere is another variation built with a similar idea.\u003c/p\u003e\n\u003ciframe scrolling=\"no\" title=\"love\" src=\"https://www.geogebra.org/material/iframe/id/azn5hcu5/width/650/height/400/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" width=\"650px\" height=\"400px\" style=\"border:0px; max-width: 100%;\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThat is the basic idea for creating a \u003cstrong\u003eLOVE\u003c/strong\u003e graph in GeoGebra. You can extend the same method to build other words or decorative shapes from equations.\u003c/p\u003e\n", "language": "en" }, { "title" : "Drawing Triangles", "date_published" : "2023-03-01T13:44:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-segitiga/", "url" : "https://www.aantriono.com/en/menggambar-segitiga/", "summary": "As we know, a triangle is a two-dimensional flat shape that has three sides and the sum of the three angles is 1800180^01800. Some time ago I posted an article about Proving the Area of ​​a Triangle\nusing GeoGebra. On this occasion we will learn how to draw triangles using GeoGebra, both equilateral triangles, isosceles triangles, arbitrary triangles, acute triangles, right triangles and obtuse triangles. Then proceed by creating a check box to show/hide the triangles. OK, let\u0026rsquo;s just look at how to draw a triangle using GeoGebra.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eAs we know, a triangle is a two-dimensional flat shape that has three sides and the sum of the three angles is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Some time ago I posted an article about \u003ca href=\"https://www.silagebra.com/2021/03/pembuktian-luas-segitiga.html\" target=\"_blank\"\u003eProving the Area of ​​a Triangle\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eusing GeoGebra. On this occasion we will learn how to draw triangles using GeoGebra, both equilateral triangles, isosceles triangles, arbitrary triangles, acute triangles, right triangles and obtuse triangles. Then proceed by creating a check box to show/hide the triangles. OK, let\u0026rsquo;s just look at how to draw a triangle using GeoGebra.\u003c/p\u003e\n\u003ch2 id=\"triangles-based-on-side-length\" class=\"headerLink\"\u003e\n \u003ca href=\"#triangles-based-on-side-length\" class=\"header-mark\" aria-label=\"Direct link to section: Triangles Based on Side Length\"\u003e\u003c/a\u003eTriangles Based on Side Length\u003c/h2\u003e\u003ch3 id=\"1-equilateral-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-equilateral-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Equilateral triangle\"\u003e\u003c/a\u003e1. Equilateral triangle\u003c/h3\u003e\u003cp\u003eThe steps for drawing an equilateral triangle are as follows.a. Open the GeoGebra application.b. Remove the axis lines/Cartesian coordinates.c. Click \u003cem\u003eCircle with Center Through Point.\u003c/em\u003e Form two circles with a radius of say 2 units. Each circle\u0026rsquo;s center is on the side of another circle.d. Click *Intersect Two Objects *to determine the intersection point of the circle.e. Click \u003cem\u003ePolygons\u003c/em\u003e to connect the three points to form a triangle. To make it clearer, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4U-lPu_ZlTS8wE-qZsilAwzjLGIeobnR5lM2HlXrntVe_mSiRA7CksCaTYqD0h3Yl3GtbTkxEW3xfGHy5mn_NneAoSHYLeHi40xSlpEVKp7GKgEmJlpPHWz_VFXq4ngGkAPo3g2KNzrw/s571/jesegitiga1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga sama sisi di geogebra\" border=\"0\" data-original-height=\"460\" data-original-width=\"571\" height=\"258\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4U-lPu_ZlTS8wE-qZsilAwzjLGIeobnR5lM2HlXrntVe_mSiRA7CksCaTYqD0h3Yl3GtbTkxEW3xfGHy5mn_NneAoSHYLeHi40xSlpEVKp7GKgEmJlpPHWz_VFXq4ngGkAPo3g2KNzrw/w320-h258/jesegitiga1.png\" title=\"segitiga sama sisi\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003ef. Click \u003cem\u003eMove\u003c/em\u003e. Remove both circles by right clicking on each circle and clicking \u003cem\u003eShow Objects\u003c/em\u003e. Next, right click on each line segment and select \u003cem\u003eObject Properties\u003c/em\u003e, click \u003cem\u003eshow label\u003c/em\u003e and select \u003cem\u003evalue\u003c/em\u003e. The result looks like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKPxN69_RUOg3bHtgSOKxSl4dJ_8mvEqW0Ec2J5ZW5dS45obbkh-SbiAKvyrOG-M7HtXojV8soTN33yXox4ccHTfvclGt7jYdURS4FJyFQYPWUBAcvdl-jMDTX3myGGAhewsEH4rvMBbE/s493/jesegitiga2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga sama sisi di geogebra\" border=\"0\" data-original-height=\"298\" data-original-width=\"493\" height=\"193\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKPxN69_RUOg3bHtgSOKxSl4dJ_8mvEqW0Ec2J5ZW5dS45obbkh-SbiAKvyrOG-M7HtXojV8soTN33yXox4ccHTfvclGt7jYdURS4FJyFQYPWUBAcvdl-jMDTX3myGGAhewsEH4rvMBbE/w320-h193/jesegitiga2.png\" title=\"segitiga sama sisi\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eMove the three corners of the triangle, what happens? Why is triangle ABC equilateral?\u003c/p\u003e\n\u003ch3 id=\"2-isosceles-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-isosceles-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Isosceles triangle\"\u003e\u003c/a\u003e2. Isosceles triangle\u003c/h3\u003e\u003cp\u003eTo draw an isosceles triangle, the steps are as follows.a. Click \u003cem\u003eCircle with Center Through Point\u003c/em\u003e, then click in two different places on the graph view.b. Click *Points *on the toolbar then click on the circle, so that the third point is obtained.c. Click \u003cem\u003ePolygons\u003c/em\u003e and connect the three points to form a triangle. Look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXOMnWZSQ2wDZojTKedpBaoo6uiTo4QCRHqTMCXtBX52poutYGuy_5ckH_lRWSVoDJlT9KeK2SQeaqtiw-K6iCL0cCzZ8fXcphbUoRO6Sl80wiPt8u7fGayU8qFbHcfidw0lE-_wR-04o/s727/jesegitiga3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga sama kaki di geogebra\" border=\"0\" data-original-height=\"425\" data-original-width=\"727\" height=\"187\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXOMnWZSQ2wDZojTKedpBaoo6uiTo4QCRHqTMCXtBX52poutYGuy_5ckH_lRWSVoDJlT9KeK2SQeaqtiw-K6iCL0cCzZ8fXcphbUoRO6Sl80wiPt8u7fGayU8qFbHcfidw0lE-_wR-04o/w320-h187/jesegitiga3.png\" title=\"segitiga sama kaki\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003ed. Click \u003cem\u003eMove\u003c/em\u003e. Remove the circle by right clicking on the circle and clicking \u003cem\u003eShow Objects\u003c/em\u003e. Next, right click on each line segment and select \u003cem\u003eObject Properties\u003c/em\u003e, click *show label *and select \u003cem\u003evalue\u003c/em\u003e. The result looks like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDNIMj6u1525c3E_w7KV8HH1YaqYmcGvaX4mSmdah7OmXoAXVzqVMxKmyWJ0TWEC3FEHhzO8iYk9doDDlosTbTxCi7oKFysJzH0ACS9Hdx5vzwmapLu7pMi-LRdFpXfi2Kc-rkgdf5ObU/s517/jesegitiga4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga sama kaki di geogebra\" border=\"0\" data-original-height=\"436\" data-original-width=\"517\" height=\"270\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDNIMj6u1525c3E_w7KV8HH1YaqYmcGvaX4mSmdah7OmXoAXVzqVMxKmyWJ0TWEC3FEHhzO8iYk9doDDlosTbTxCi7oKFysJzH0ACS9Hdx5vzwmapLu7pMi-LRdFpXfi2Kc-rkgdf5ObU/w320-h270/jesegitiga4.png\" title=\"segitiga sama kaki\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eMove the three corners of the triangle, what happens? Why is triangle EFG isosceles?\u003c/p\u003e\n\u003ch3 id=\"3-any-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-any-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Any triangle\"\u003e\u003c/a\u003e3. Any triangle\u003c/h3\u003e\u003cp\u003eTo draw any triangle, the steps are as follows.a. Click \u003cem\u003ePolygons\u003c/em\u003e and any triangle shape provided that the three sides are not the same length.b. Click \u003cem\u003eMove\u003c/em\u003e. Next, right click on each line segment and select \u003cem\u003eObject Properties\u003c/em\u003e, click \u003cem\u003eshow label\u003c/em\u003e and select \u003cem\u003evalue.\u003c/em\u003e The result looks like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp-DD2rFD-EBB98Hw9PjAErji13mjpoaf9nCEjWMjg-FFKskFFPb5LHR0wYvtm3KwRl3AzB0Vik-myCXQi1L0tU9blU0pQ7P6KeuG-gOXBPC9E3ViR-YfLSovkLYtetnk7pFMjwPCrYEo/s509/jesegitiga5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga sembarang di geogebra\" border=\"0\" data-original-height=\"414\" data-original-width=\"509\" height=\"260\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp-DD2rFD-EBB98Hw9PjAErji13mjpoaf9nCEjWMjg-FFKskFFPb5LHR0wYvtm3KwRl3AzB0Vik-myCXQi1L0tU9blU0pQ7P6KeuG-gOXBPC9E3ViR-YfLSovkLYtetnk7pFMjwPCrYEo/w320-h260/jesegitiga5.png\" title=\"segitiga sembarang\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eMove the three corners of the triangle, what happens? Why is triangle HIJ called an arbitrary triangle?\u003c/p\u003e\n\u003cdiv class=\"bacajuga\"\u003e\n\u003ch4 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also Read\"\u003e\u003c/a\u003eAlso Read\u003c/h4\u003e\u003cul\u003e \u003cli\u003e\u003ca href=\"https://www.silagebra.com/2021/03/pembuktian-luas-segitiga.html\"\u003eProving the Area of ​​a Triangle\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.silagebra.com/2021/03/animasi-jam.html\"\u003eClock Animation\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.silagebra.com/2021/03/bayangan-dua-lingkaran.html\"\u003eShadow of Two Circles\u003c/a\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/div\u003e\n\u003ch2 id=\"triangles-based-on-angle-measures\" class=\"headerLink\"\u003e\n \u003ca href=\"#triangles-based-on-angle-measures\" class=\"header-mark\" aria-label=\"Direct link to section: Triangles Based on Angle Measures\"\u003e\u003c/a\u003eTriangles Based on Angle Measures\u003c/h2\u003e\u003ch3 id=\"1-acute-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-acute-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Acute triangle\"\u003e\u003c/a\u003e1. Acute triangle\u003c/h3\u003e\u003cp\u003eThe steps for drawing an acute triangle are as follows.a. Click *Polygons *and form a triangle with all angles less than \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.b. Click \u003cem\u003eMove\u003c/em\u003e. Right click on each line and click *Show Labels.*c. Click \u003cem\u003eAngle\u003c/em\u003e. Click on the triangle. To make it clearer, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtlJMvUMtORF-oRPqlvVOBLnWUf8xz3fva16D2W9z8wsvopoRcDU6z6a9nieLhBMKxTLrIoVUD0K7UB6fEGp8Z7Q3TMWJMYZzctZ6BZ8BVfE05l0nM4E3tldpjHBrKruGDwaEpqdlUpb4/s509/jesegitiga6.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga lancip di geogebra\" border=\"0\" data-original-height=\"279\" data-original-width=\"509\" height=\"175\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtlJMvUMtORF-oRPqlvVOBLnWUf8xz3fva16D2W9z8wsvopoRcDU6z6a9nieLhBMKxTLrIoVUD0K7UB6fEGp8Z7Q3TMWJMYZzctZ6BZ8BVfE05l0nM4E3tldpjHBrKruGDwaEpqdlUpb4/w320-h175/jesegitiga6.png\" title=\"segitiga lancip\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eWhy KLM triangle is called an acute triangle?\u003c/p\u003e\n\u003ch3 id=\"2-right-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-right-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Right triangle\"\u003e\u003c/a\u003e2. Right triangle\u003c/h3\u003e\u003cp\u003eThe steps for drawing a right triangle are as follows.a. Click \u003cem\u003eSegment\u003c/em\u003e. Click on the graphic display so that a horizontal line formsb. Click \u003cem\u003ePerpendicular Line\u003c/em\u003e. Click the line in step a and place the line formed at the end of the perpendicular line. c. Click \u003cem\u003ePoints\u003c/em\u003e. Click on the vertical line formed in step b.d. Click \u003cem\u003ePolygons\u003c/em\u003e and connect the three points to form a triangle. To make it clearer, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtvv_W3ArZmBNQNsSkaqQWw0njhQAu2678cI3JyrYyHp0tej9JiB7H6vU0mwfTrxueAdiwWNjI5EP3STvc_bEJR2XraeFK4tbCkpKx5YwjvgeZ9x6sXdMs-JDHWDYliUuFO0YBRLE1_As/s514/jesegitiga7.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga siku-siku di geogebra\" border=\"0\" data-original-height=\"319\" data-original-width=\"514\" height=\"199\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtvv_W3ArZmBNQNsSkaqQWw0njhQAu2678cI3JyrYyHp0tej9JiB7H6vU0mwfTrxueAdiwWNjI5EP3STvc_bEJR2XraeFK4tbCkpKx5YwjvgeZ9x6sXdMs-JDHWDYliUuFO0YBRLE1_As/w320-h199/jesegitiga7.png\" title=\"segitiga siku-siku\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003ee. Click \u003cem\u003eMove\u003c/em\u003e. Right click on each line and click \u003cem\u003eShow Labels\u003c/em\u003e.f. Click \u003cem\u003eAngle\u003c/em\u003e. Click on the triangle. To make it clearer, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRuUHNQbpAC1a0XTyjtjvkt64GRVVit_cuptyoielvQdt1zRui_ga6tf-dAGb4OAlb4L6NImTCPSnptYBmjfOOJgSGoYahtS00MaTgNuRVqIIhLNuGK-W9wc1Tx9i15YIXyirC6ABxoh8/s512/jesegitiga8.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga siku-siku di geogebra\" border=\"0\" data-original-height=\"326\" data-original-width=\"512\" height=\"204\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRuUHNQbpAC1a0XTyjtjvkt64GRVVit_cuptyoielvQdt1zRui_ga6tf-dAGb4OAlb4L6NImTCPSnptYBmjfOOJgSGoYahtS00MaTgNuRVqIIhLNuGK-W9wc1Tx9i15YIXyirC6ABxoh8/w320-h204/jesegitiga8.png\" title=\"segitiga siku-siku\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eWhy is the PNO triangle called a right triangle?\u003c/p\u003e\n\u003ch3 id=\"3-obtuse-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-obtuse-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Obtuse triangle\"\u003e\u003c/a\u003e3. Obtuse triangle\u003c/h3\u003e\u003cp\u003eThe steps for drawing an obtuse triangle are as follows.a. Click \u003cem\u003ePolygons\u003c/em\u003e and form a triangle with one corner more than \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.b. Click \u003cem\u003eMove\u003c/em\u003e. Right click on each line and click *Show Labels.*c. Click \u003cem\u003eAngle\u003c/em\u003e. Click on the triangle. To make it clearer, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrjhufakhsdLqLBqXEJpdBdMaSxWrzibSTOFY4zCzY8H8y8KVEPf-XjkmqlDJwKsftows1b0O1skSfjVjKBBg-zdeuMwH7LFsR67DmILWZySqkDav2M9L-p3wssCrUSnkKWFpibWD_Nug/s490/jesegitiga9.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"segitiga tumpul di geogebra\" border=\"0\" data-original-height=\"337\" data-original-width=\"490\" height=\"220\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrjhufakhsdLqLBqXEJpdBdMaSxWrzibSTOFY4zCzY8H8y8KVEPf-XjkmqlDJwKsftows1b0O1skSfjVjKBBg-zdeuMwH7LFsR67DmILWZySqkDav2M9L-p3wssCrUSnkKWFpibWD_Nug/w320-h220/jesegitiga9.png\" title=\"segitiga tumpul\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eWhy is the QRS triangle called an arbitrary triangle?\u003c/p\u003e\n\u003ch2 id=\"coloring-and-caption-text\" class=\"headerLink\"\u003e\n \u003ca href=\"#coloring-and-caption-text\" class=\"header-mark\" aria-label=\"Direct link to section: Coloring and Caption Text\"\u003e\u003c/a\u003eColoring and Caption Text\u003c/h2\u003e\u003cp\u003eNext, so that the appearance of the triangle that has been drawn looks more attractive, give it a good coloring by right clicking on the triangle, selecting it \u003cem\u003eObject Properties\u003c/em\u003e, choose \u003cem\u003ecolour\u003c/em\u003e. Then add text as additional information to each triangle by clicking \u003cem\u003eText\u003c/em\u003e and write a description of the triangle. Next click \u003cem\u003eOK\u003c/em\u003e. To make it clearer, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCnueDaRLkbCZ07oUH-Oh-0LBmXfNGYTy7iJwxxwFvd4KftSGvaMkiAWC85fRLB9WAalY77_yxn-CVUEkiPmKNsxu0qLjnOX9VACJFHAwbQB7NgGnxhzaMO49T5s_30zCmZYzoYkBuGks/s645/jesegitiga10.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Jenis-jenis segitiga\" border=\"0\" data-original-height=\"393\" data-original-width=\"645\" height=\"195\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCnueDaRLkbCZ07oUH-Oh-0LBmXfNGYTy7iJwxxwFvd4KftSGvaMkiAWC85fRLB9WAalY77_yxn-CVUEkiPmKNsxu0qLjnOX9VACJFHAwbQB7NgGnxhzaMO49T5s_30zCmZYzoYkBuGks/w320-h195/jesegitiga10.png\" title=\"Jenis-jenis segitiga\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"create-check-boxes\" class=\"headerLink\"\u003e\n \u003ca href=\"#create-check-boxes\" class=\"header-mark\" aria-label=\"Direct link to section: Create Check Boxes\"\u003e\u003c/a\u003eCreate Check Boxes\u003c/h2\u003e\u003cp\u003ePreviously, we created two texts with the titles \u0026ldquo;Triangles based on side lengths\u0026rdquo; and \u0026ldquo;Triangles based on angle sizes\u0026rdquo;.\u003c/p\u003e\n\u003ch3 id=\"1-equilateral-triangle-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-equilateral-triangle-1\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Equilateral triangle\"\u003e\u003c/a\u003e1. Equilateral triangle\u003c/h3\u003e\u003cp\u003eTo make \u003cem\u003echeck box\u003c/em\u003e equilateral triangle the steps are as follows.a. Click \u003cem\u003eCheck Box\u003c/em\u003e.b. In column \u003cem\u003eCaptions\u003c/em\u003e type equilateral trianglec. Click the small arrow below the text \u003cem\u003eSelect objects in construction or choose from list\u003c/em\u003e. Next, click point A, point B, point C, segment AB, segment BC, segment AC, triangle caption text ABC, and polygon ABC. Next click \u003cem\u003eApply\u003c/em\u003e. To make it clearer, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNU6TYZPDiV0xVH06TrniTzgoGrjmcImUXrH7zqgdr5eYek7EdpjB2A-AyJN1KblAL5pX3piDG2-0AOBCGeHRkXJeUEGCprfMBfbY-CoERaspzUJgDiDLFzqya8RIpbAJsU6QzUJCJj0s/s561/jesegitiga11.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"check box di geogebra\" border=\"0\" data-original-height=\"439\" data-original-width=\"561\" height=\"250\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNU6TYZPDiV0xVH06TrniTzgoGrjmcImUXrH7zqgdr5eYek7EdpjB2A-AyJN1KblAL5pX3piDG2-0AOBCGeHRkXJeUEGCprfMBfbY-CoERaspzUJgDiDLFzqya8RIpbAJsU6QzUJCJj0s/w320-h250/jesegitiga11.png\" title=\"check box di geogebra\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"2-isosceles-triangle-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-isosceles-triangle-1\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Isosceles triangle\"\u003e\u003c/a\u003e2. Isosceles triangle\u003c/h3\u003e\u003cp\u003eTo make \u003cem\u003echeck box\u003c/em\u003e isosceles triangle the steps are as follows.a. Click *Check Box.*b. In column \u003cem\u003eCaptions\u003c/em\u003e type isosceles trianglec. Click the small arrow below the text \u003cem\u003eSelect objects in construction or choose from list\u003c/em\u003e. Next, click point E, point F, point G, segment EF, segment FG, segment EG, triangle text EFG, and polygon EFG. Next click \u003cem\u003eApply.\u003c/em\u003e\u003c/p\u003e\n\u003ch3 id=\"3-any-triangle-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-any-triangle-1\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Any triangle\"\u003e\u003c/a\u003e3. Any triangle\u003c/h3\u003e\u003cp\u003eTo create any triangle check box, the steps are as follows.a. Click \u003cem\u003eCheck Box\u003c/em\u003e.b. In column \u003cem\u003eCaptions\u003c/em\u003e type any trianglec. Click the small arrow below the text* Select objects in construction or choose from list*. Next, click point H, point I, point J, segment HI, segment HJ, segment IJ, triangle text HIJ, and polygon HIJ. Next click *Apply. *\u003c/p\u003e\n\u003ch3 id=\"3-acute-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-acute-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Acute triangle\"\u003e\u003c/a\u003e3. Acute triangle\u003c/h3\u003e\u003cp\u003eTo make \u003cem\u003echeck box\u003c/em\u003e acute triangle the steps are as follows.a. Click *Check Box.*b. In column \u003cem\u003eCaptions\u003c/em\u003e type acute triangle.c. Click the small arrow below the text \u003cem\u003eSelect objects in construction or choose from list\u003c/em\u003e. Next, click point K, point L, point M, segment Kl, segment KM, segment LM, angles of the KLM triangle, text description of the KLM triangle, and KLM polygon. Next click *Apply. *\u003c/p\u003e\n\u003ch3 id=\"4-right-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-right-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Right triangle\"\u003e\u003c/a\u003e4. Right triangle\u003c/h3\u003e\u003cp\u003eTo make \u003cem\u003echeck box\u003c/em\u003e right triangle the steps are as follows.a. Click Check Box.b. In column *Captions *type right triangle.c. Click the small arrow below the text \u003cem\u003eSelect objects in construction or choose from list\u003c/em\u003e. Next, click point P, point N, point O, segment PN, segment NO, segment OP, triangle angle PNO, triangle description text PNO, and polygon PNO. Next click *Apply. *\u003c/p\u003e\n\u003ch3 id=\"6-obtuse-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-obtuse-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Obtuse triangle\"\u003e\u003c/a\u003e6. Obtuse triangle\u003c/h3\u003e\u003cp\u003eTo make *check box *obtuse triangle the steps are as follows.a. Click \u003cem\u003eCheck Box\u003c/em\u003e.b. In column \u003cem\u003eCaptions\u003c/em\u003e type obtuse triangle.c. Click the small arrow below the text \u003cem\u003eSelect objects in construction or choose from list\u003c/em\u003e. Next, click point Q, point R, point S, QR segment, RS segment, QS segment, QRS triangle angle, QRS triangle description text, and QRS polygon. Next click *Apply. *The result of carrying out these steps looks like following image. \u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3iResfBAASQXIMSL9e0qr8VwNQUpm232mCp17xmIzhYWZyixkTXcrIw_UszDZZyVWHV1vJahzM_keoRZMvmfjIVtSGa88_ZzW-11pwIFG7WBqXc-XeJHaVCFYwl0IKilZ7hoU-xbXXHs/s637/jesegitiga12.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"jenis-jenis segitiga\" border=\"0\" data-original-height=\"507\" data-original-width=\"637\" height=\"255\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3iResfBAASQXIMSL9e0qr8VwNQUpm232mCp17xmIzhYWZyixkTXcrIw_UszDZZyVWHV1vJahzM_keoRZMvmfjIVtSGa88_ZzW-11pwIFG7WBqXc-XeJHaVCFYwl0IKilZ7hoU-xbXXHs/w320-h255/jesegitiga12.png\" title=\"jenis-jenis segitiga\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eTo check the results, please check/uncheck each box/check box that has been created and pay attention to the triangle.\u003c/p\u003e\n\u003ch2 id=\"display-on-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#display-on-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: Display on GeoGebra\"\u003e\u003c/a\u003eDisplay on GeoGebra\u003c/h2\u003e\u003cp\u003eBelow is an example of the results of drawing a triangle using GeoGebra. Please move the corner point and click each triangular box at the bottom.\u003c/p\u003e\n\u003ciframe height=\"541px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/bsgrut45/width/650/height/541/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Jnis Segitiga\" width=\"650px\"\u003e \u003c/iframe\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eBy drawing triangles using GeoGebra, of course we can more easily convey learning to students about triangle material. Moreover, if the image is given additional animation effects, it will definitely be more interesting and easier to understand. Of course, there are still many benefits of GeoGebra in learning mathematics (geometry, calculus, algebra) which we can learn with regular practice. That\u0026rsquo;s the article on drawing triangles using GeoGebra. If there are errors, please correct them, and if you have input, please write them here. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Parallelogram Animation", "date_published" : "2023-03-01T13:44:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/animasi-jajar-genjang/", "url" : "https://www.aantriono.com/en/animasi-jajar-genjang/", "summary": "JA parallelogram or parallelogram is a two-dimensional flat shape formed by two pairs of edges, each of which is the same length and parallel to its partner, and has two pairs of angles, each of which is the same size as the angle opposite it.\nFormula for the perimeter and area of ​​a parallelogramThe perimeter and area of ​​a parallelogram can be found using the following formula.Around=Sum of all sides=2×base+2×hypotenuse=2×(base + hypotenuse)Area=base×height=a×t\\begin{aligned} \\text{Around}\u0026amp;=\\text{Sum of all sides}\\\\ \u0026amp;=2\\times \\text{base} + 2\\times \\text{hypotenuse}\\\\ \u0026amp;=2\\times (\\text{base + hypotenuse})\\\\[12pt] \\text{Area}\u0026amp;=\\text{base}\\times \\text{height}\\\\ \u0026amp;=a\\times t \\end{aligned}AroundArea​=Sum of all sides=2×base+2×hypotenuse=2×(base + hypotenuse)=base×height=a×t​\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eJA parallelogram or parallelogram is a two-dimensional flat shape formed by two pairs of edges, each of which is the same length and parallel to its partner, and has two pairs of angles, each of which is the same size as the angle opposite it.\u003c/p\u003e\n\u003ch2 id=\"formula-for-the-perimeter-and-area-of-a-parallelogram\" class=\"headerLink\"\u003e\n \u003ca href=\"#formula-for-the-perimeter-and-area-of-a-parallelogram\" class=\"header-mark\" aria-label=\"Direct link to section: Formula for the perimeter and area of ​​a parallelogram\"\u003e\u003c/a\u003eFormula for the perimeter and area of ​​a parallelogram\u003c/h2\u003e\u003cp\u003eThe perimeter and area of ​​a parallelogram can be found using the following formula.\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eAround\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmtext\u003eSum of all sides\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003ebase\u003c/mtext\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003ehypotenuse\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmtext\u003ebase + hypotenuse\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eArea\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmtext\u003ebase\u003c/mtext\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003eheight\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{aligned} \\text{Around}\u0026amp;=\\text{Sum of all sides}\\\\ \u0026amp;=2\\times \\text{base} + 2\\times \\text{hypotenuse}\\\\ \u0026amp;=2\\times (\\text{base + hypotenuse})\\\\[12pt] \\text{Area}\u0026amp;=\\text{base}\\times \\text{height}\\\\ \u0026amp;=a\\times t \\end{aligned}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:8.7em;vertical-align:-4.1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.6em;\"\u003e\u003cspan style=\"top:-6.76em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eAround\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.26em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.76em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.06em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eArea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.44em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.1em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.6em;\"\u003e\u003cspan style=\"top:-6.76em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eSum of all sides\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.26em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ehypotenuse\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.76em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase + hypotenuse\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.06em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eheight\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.44em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.1em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003ch2 id=\"draw-a-parallelogram\" class=\"headerLink\"\u003e\n \u003ca href=\"#draw-a-parallelogram\" class=\"header-mark\" aria-label=\"Direct link to section: Draw a parallelogram\"\u003e\u003c/a\u003eDraw a parallelogram\u003c/h2\u003e\u003cp\u003eTo draw a parallelogram, we can use the GeoGebra application, then also create an animation so it will be easier to find out the perimeter and area. The method for drawing a parallelogram using GeoGebra is as follows.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eOpen the GeoGebra application.\u003c/li\u003e\n\u003cli\u003eEliminate the Cartesian line.\u003c/li\u003e\n\u003cli\u003eOn \u003cem\u003etoolbars\u003c/em\u003e, choose \u003cem\u003eCircle with Center Through Point\u003c/em\u003e. Next click on the graph screen.\u003c/li\u003e\n\u003cli\u003eStill on \u003cem\u003etoolbars\u003c/em\u003e, choose \u003cem\u003eLine\u003c/em\u003e. Click on both points so they connect with another line through the center point of the circle.\u003c/li\u003e\n\u003cli\u003eOn \u003cem\u003etoolbars\u003c/em\u003e, choose \u003cem\u003eparallel lines\u003c/em\u003e. Draw a line* h* parallel *f *and lines \u003cem\u003ei\u003c/em\u003e parallel \u003cem\u003eg\u003c/em\u003e. Look at the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_a76s9LwDnlBBHwOBrPDgF_de_zp0h5MNx92eEtlakXlU9-J7Df36AtyFkDr-fs0EkeOZ-e8HcNv6BkUzHsiD3C5uQ3cw5xyQk915-tyUwB9BD3PFP88zu5JZN-lJq93I4Om7AKGYr6s/w320-h192/jajargenjang2.png' alt=\"Parallelogram in GeoGebra\" \u003e\u003c/li\u003e\n\u003cli\u003eOn the menu \u003cem\u003etoolbars\u003c/em\u003e, click \u003cem\u003eIntersect\u003c/em\u003e, click the line \u003cem\u003eh\u003c/em\u003e And \u003cem\u003ei\u003c/em\u003e. Next click \u003cem\u003ePolygons\u003c/em\u003e. Create a parallelogram and color it accordingly. Look at the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBXjrrMt07gTFTH-LLRy3PhLt4n5-VfTxWu0lXk0qzSk7bT1p_DVzu626LdSdaQPO9UCX9GI2jPU8ht9jav4ZG3soQqJnDy8Ds8TqmhWv2nA9Kfw6rV2rEBTiBr-XShWnHS1qBedzP6TU/w320-h202/jajargenjang3.png' alt=\"Parallelogram in GeoGebra\" \u003e\u003c/li\u003e\n\u003cli\u003eOn \u003cem\u003etoolbars\u003c/em\u003e, click \u003cem\u003eMove\u003c/em\u003e. Right click on each line and select \u003cem\u003eShow Objects\u003c/em\u003e. Look at the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwmV_PIWpEie9xee9-UZlqImAmgQqohSepFuTgLo54t0IzSwNEknbPKVa2OWRAICseFDt1hv8hpeJNOaW93VNAEHSVtUpp7TxCznydxSkI1H_3HFdEwPsKcKK-GFiccOX85vuTedeHsFg/w320-h205/jajargenjang4.png' alt=\"Parallelogram in GeoGebra\" \u003e\u003c/li\u003e\n\u003cli\u003eAfter all the lines have been removed, the parallelogram image looks like the following.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJWUmSamQ_RFlUCno3boisOLdUZS92r5nDU-NL3Ymh3kBXv-QJTKy7BuZB0DhYOAkrv2fiN4_9Ay1HlMhe_5lb7vxVl0Jd0og6HbOOllaWMsLYL6addUNWtaPOdqL4sxzj5dPsSB2NVUE/w320-h198/jajargenjang5.png' alt=\"Parallelogram in GeoGebra\" \u003e\u003c/li\u003e\n\u003cli\u003eNext to display the angle, click \u003cem\u003eAngle\u003c/em\u003e on \u003cem\u003etoolbars\u003c/em\u003e and click on the parallelogram area.\u003c/li\u003e\n\u003cli\u003eTo display the side lengths and perimeter of the parallelogram, click *Distance or Length *on \u003cem\u003etoolbars\u003c/em\u003e, then click on each side of the parallelogram and click on the parallelogram area.\u003c/li\u003e\n\u003cli\u003eTo display the area of ​​the parallelogram, click \u003cem\u003eAreas\u003c/em\u003e on the toolbar, then click on the parallelogram area. The entirety of step 9 to step 11 looks like the following image.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjR_aLRaFx7a7Yq_YuGzDmgT-R8ANW5iph42-UebE3Hkh_eRV2deaaFRSXzmDmY5t3qKRyLj_YEAt1jJlF5G4c5tdYDVHDI9c7lxVWwy_iB3Kn2TW84i7kXP93EbbqF_ytCdGUhA4P4BtQ/w320-h256/jajargenjang6.png' alt=\"Parallelogram in GeoGebra\" \u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"find-the-perimeter-of-a-parallelogram\" class=\"headerLink\"\u003e\n \u003ca href=\"#find-the-perimeter-of-a-parallelogram\" class=\"header-mark\" aria-label=\"Direct link to section: Find the perimeter of a parallelogram\"\u003e\u003c/a\u003eFind the perimeter of a parallelogram\u003c/h2\u003e\u003cp\u003eTo find the perimeter of a parallelogram, we can calculate the length of the base side AC, slant side CD, top side BD, and slant side AD, then add up the lengths of the four sides. Below is a simulation to find the circumference of a parallelogram. Move point A, B, C, or D to change the shape of the parallelogram. Note changes in side lengths, perimeters, and angles.\u003c/p\u003e\n\u003ciframe height=\"450px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/rpdd34av/width/650/height/450/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"jajaran genjang\" width=\"650px\"\u003e \u003c/iframe\u003e\n\u003ch2 id=\"find-the-area-of-a-parallelogram\" class=\"headerLink\"\u003e\n \u003ca href=\"#find-the-area-of-a-parallelogram\" class=\"header-mark\" aria-label=\"Direct link to section: Find the area of ​​a parallelogram\"\u003e\u003c/a\u003eFind the area of ​​a parallelogram\u003c/h2\u003e\u003cp\u003eTo find the formula for the area of ​​a parallelogram, we can shift the parallelogram part, it turns out that after shifting it forms a rectangle. The base of the parallelogram is equal to the length of the rectangle and the height of the parallelogram is equal to the width of the rectangle. To make it clearer, pay attention to the following GeoGebra simulation. Please move the slider to the right, or click the OK button, and to return to its original form, please click the back button.\u003c/p\u003e\n\u003ciframe height=\"450px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/zmge28mr/width/650/height/450/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Jajar Genjang\" width=\"650px\"\u003e \u003c/iframe\u003e\nTo create an animation of finding the area of ​​a parallelogram as above, the steps are as follows.\n1. Open the GeoGebra application.\n2. Eliminate the Cartesian line.\n3. On *toolbars*, choose *Polygons*. Create a triangle and trapezoid to form a parallelogram as follows.\n![Parallelogram in GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNEI196Kju9r-4Ko9OKNb9Ysbz_GUzm1qNJ1oP2sTAuQ7N8T_AkQUFcCZr9mcmJ9JxAYii4cHfUTSmQprCpB0WO6CA8Ts84gjumLHKgYAIIFD35lQTQg_MZpsy9lUi5jtO3E7K2CugMPY/w320-h160/jajargenjang7.png)\n4. On *toolbars*, choose *Sliders*. Enter 0 in Min, 6 in Max, and 1 in Increment. Next click *OK*. Look at the following image.\n![Slider in GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkyfOOcgdEHnr-NYLv9Gbxd919zkkIlOU6wcNv-Gx8VEK3XPJAKRlFUSdP4qwmqANYxe0VWBn89s0W_ayeYQPuhoyV9ccF66DMwsx7_iftxtCG49FXdUZWPURpc7DD2Vnax_-ySL4BY2M/w320-h183/jajargenjang8.png)\n5. On *toolbars*, *click Segment with Given Length*. Click the points to be shifted (A, B, and E) as far as the slider \"*shift*\". Click *Segment with Given Length *, click point A, write shift. Likewise for points B and E. Look at the following picture.\n![Parallelogram in GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtfkk8q1O2zCCD_KUUxgocSPP7Mnsxrxbas2DT_Tm4yIaCs7dMpUoyYpEO9Z9U60gdDwyCq8ZahKsH87jpLkWur-3zdAdMkURA7SJ72En9BgwIPr1FF7xfe9X47j2SExL2g7fSqAp3dLg/w320-h148/jajargenjang9.png)\n6. To create an animated button, click on the toolbar *Buttons*, then click on the screen, fill in *OK* on *Captions,* * StartAnimation[slide]* on Geogebra Script. Click *OK.* Look at the following image.\n![OK button in GeoGebra](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvjVrCqZg3dTu-clxlKUHpihMblIlrx2FjcFdf4dNEyKNpdVSBWNSs4AgIk0H6i849LIxmtG0p_BLndLb6FyrbeLk5A-ryFuNsTuIkyrWBa5kFjT0cKSJ7Yb1Z2wJpVbFvANTBnweevTU/w320-h211/jajargenjang10.png)\nTo create a button *Return*, fill in *StartAnimation[false] slide=0* on *GeoGebra Script.*\n\u003col start=\"7\"\u003e\n\u003cli\u003eNext to speed up the movement \u003cem\u003eSliders,\u003c/em\u003e Right click on the slider, a dialog box appears as follows.\n\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHDRcxffe7gIPa3LtDZkquQDJeHvmLdcsrfmmoXmt9QGufv7TybPc1etDrj9Lu68qRTzrlWCTePV_QwOQSeA7lTmrIsyiHrjnHObRusNF8W6E2yirWexxVDTR5oqG3RpmqUObrQu1PtR0/w200-h320/jajargenjang11.png' alt=\"Slider in GeoGebra\" \u003e\nEnter numbers in \u003cem\u003eSpeed\u003c/em\u003e, choose \u003cem\u003eIncreasing (Once)\u003c/em\u003e so that the movement only goes up or forward. Add color to \u003cem\u003eSliders\u003c/em\u003e, knob \u003cem\u003eOK\u003c/em\u003e And \u003cem\u003eReturn\u003c/em\u003e as desired.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eBy using GeoGebra, we can draw, create animations and then also find the perimeter and area of ​​the parallelogram. Of course, this is very helpful for teachers in teaching parallelogram material to their students. That\u0026rsquo;s the article about parallelograms, if there are errors, please correct them, if you have input, please write them here. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Drawing a Clock in LaTeX", "date_published" : "2023-03-01T13:43:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-jam-di-latex/", "url" : "https://www.aantriono.com/en/menggambar-jam-di-latex/", "summary": " IntroductionIn an earlier post, the blog covered how to create a clock animation in GeoGebra. This article focuses on a different approach: drawing an analog clock directly in LaTeX with TikZ.\nThe final result will look like this.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn an earlier post, the blog covered how to create a \u003ca href=\"https://www.silagebra.com/2021/03/animasi-jam.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eclock animation in GeoGebra\u003c/a\u003e. This article focuses on a different approach: drawing an analog clock directly in LaTeX with TikZ.\u003c/p\u003e\n\u003cp\u003eThe final result will look like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhelTQ1EFMIKr0msDakKxUjgBAfkfkC7b8DyU9t40j1ol7qzn84-6OjRzXI1efwAoP5Do1Yu0pSAOTYZiEZibyY4wSzELMEeo70AyJuHHE3d4qxz9Blae_L-a4C4IhG8mVfzmQVev0233c/s407/jam.png' alt=\"Example of a clock drawn in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"step-by-step\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step\" class=\"header-mark\" aria-label=\"Direct link to section: Step by Step\"\u003e\u003c/a\u003eStep by Step\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eOpen your LaTeX editor, either online or offline.\u003c/li\u003e\n\u003cli\u003eLoad the \u003ccode\u003etikz\u003c/code\u003e and \u003ccode\u003excolor\u003c/code\u003e packages in the preamble.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eDefine the drawing scale. This example uses a scale of \u003ccode\u003e1.25\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[scale=1.25]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eDraw the main circle of the clock.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,0) circle(0.9);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eYou can change the color or adjust the radius \u003ccode\u003e0.9\u003c/code\u003e to fit your design.\u003c/p\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eAdd the hour numbers around the circle.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\angle\u003c/span\u003e / \u003cspan class=\"k\"\u003e\\label\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0/3, 30/2, 60/1, 90/12, 120/11, 150/10, 180/9, 210/8, 240/7, 270/6, 300/5, 330/4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\angle\u003c/span\u003e:0.75) node[text=white]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tiny\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\label\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eA full circle is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, so each hour marker is separated by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e30^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eDraw the hour hand.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=1.5pt, white, cap=round]\u003c/span\u003e (0,0) -- (118:0.3);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eDraw the minute hand.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=1.5pt, white, cap=round]\u003c/span\u003e (0,0) -- (0:0.5);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"9\"\u003e\n\u003cli\u003eDraw the second hand.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=.5pt, red, cap=round]\u003c/span\u003e (0,0) -- (-120:0.6);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eYou can change the position, length, color, and thickness of each hand to show a different time.\u003c/p\u003e\n\u003col start=\"10\"\u003e\n\u003cli\u003eAdd a small circle at the center of the clock.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue]\u003c/span\u003e (0,0) circle(1.2pt);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"11\"\u003e\n\u003cli\u003eIf you want, add a small brand label or name inside the clock face.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.5]\u003c/span\u003e at (90:0.6) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewhite\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tiny\u003c/span\u003e AAN\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.5]\u003c/span\u003e at (-90:0.6) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewhite\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tiny\u003c/span\u003e TRIONO\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"12\"\u003e\n\u003cli\u003eClose the \u003ccode\u003etikzpicture\u003c/code\u003e environment and the document.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"full-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#full-code\" class=\"header-mark\" aria-label=\"Direct link to section: Full Code\"\u003e\u003c/a\u003eFull Code\u003c/h2\u003e\u003cp\u003eHere is the complete example.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eClock\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAan Triono\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan 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class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,0) circle(0.9);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=1.5pt, white, cap=round]\u003c/span\u003e (0,0) -- (118:0.3);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=1.5pt, white, cap=round]\u003c/span\u003e (0,0) -- (0:0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[line width=.5pt, red, cap=round]\u003c/span\u003e (0,0) -- (-120:0.6);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan 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class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[scale=0.5]\u003c/span\u003e at (-90:0.6) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\color\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewhite\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tiny\u003c/span\u003e TRIONO\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eTikZ makes it possible to draw simple objects like an analog clock directly in LaTeX without relying on another application. For more ideas and references, see \u003ca href=\"https://www.latexdraw.com/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003elatexdraw.com\u003c/a\u003e.\u003c/p\u003e\n", "language": "en" }, { "title" : "How to Draw Yin and Yang Symbols", "date_published" : "2023-03-01T13:43:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-menggambar-simbol-yin-dan-yang/", "url" : "https://www.aantriono.com/en/cara-menggambar-simbol-yin-dan-yang/", "summary": " Drawing Yin and Yang Symbol in LaTeXFollowing from wikipedia, Yin and Yang are concepts in Chinese philosophy that usually describe the nature of the interconnected and opposing forces in this world and how they mutually build on each other. On this occasion, I will not discuss the meaning of Yin and Yang in life, but will discuss how to draw the Yin and Yang symbols using LaTeX. ", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-yin-and-yang-symbol-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-yin-and-yang-symbol-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Yin and Yang Symbol in LaTeX\"\u003e\u003c/a\u003eDrawing Yin and Yang Symbol in LaTeX\u003c/h2\u003e\u003cp\u003eFollowing from wikipedia, Yin and Yang are concepts in Chinese philosophy that usually describe the nature of the interconnected and opposing forces in this world and how they mutually build on each other. On this occasion, I will not discuss the meaning of Yin and Yang in life, but will discuss how to draw the Yin and Yang symbols using LaTeX. \u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTqxnB3Y165pxBoZLypdg_YUrQryhOCCET6CI1-olxHcPhn2f9DFgyqeurbPnvQPadewu5K96-JUNlsj1EsIn42Tvx1SeXybl2z4ggZK0ZCuIrHkYHojKpjpNOY0yhXPfV27zEWM-znss/s16000/Downloads.png' alt=\"simbol Yin dan Yang\" \u003e\u003c/p\u003e\n\u003c!--more--\u003e\n\u003ch2 id=\"step-by-step-tikz-construction\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-tikz-construction\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step TikZ Construction\"\u003e\u003c/a\u003eStep-by-Step TikZ Construction\u003c/h2\u003e\u003cp\u003eOK, let\u0026rsquo;s go straight to the text editor, either online or offline.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen your text editor.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eCreate a new project.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eAs usual, to draw, you need a TikZ package and include it in the preamble.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eNext, in the body of the document, write the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eThen we will create a circle with a radius of 1 cm using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) circle (1cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eAs seen in the picture above there are black and white parts. We will divide the circle formed into two parts, each black and white using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\clip\u003c/span\u003e (0,0) circle (1cm); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0cm,1cm) rectangle (-1cm, -1cm); \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe results of these steps look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQeM0rKOQGybNCIEJstpNYNPcapLSKNaBxOfmzFEBx3V9RhJbh8K2q5rEg1Ghr-VHtWpufUyMqFMI6jsDi7qDxqr_wcBw6Wx_Z6yk8u5xDcQyw8lkavHN2SYawiQOnE4CS4dpse1jZ9bk/w320-h116/Screenshot+2021-06-09+11.06.01.png' alt=\"Simbol Yin dan yang di latex\" \u003e\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eNext we will draw the head using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,0.5) circle (0.5cm); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (0,-0.5) circle (0.5cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe results of these steps look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjL4dAYepIOH_pz_8NEjAmmvp8y2zcnC4lsX5r9PZ5sHnbhn5g2qK7EPycOmNAIKUA-Fc1RC0jh1BzrSdHNvYVhRJgn0El6XEzqGoEhm7b2RXMlqP-b0A9h9IA-Q2islQfEyZUlCZGzfpU/w320-h139/Screenshot+2021-06-09+11.09.40.png' alt=\"simbol yin dan yang di latex\" \u003e\u003c/p\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eNext we will draw the eyes using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (0,0.5) circle (0.1cm); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[black]\u003c/span\u003e (0,-0.5) circle (0.1cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"9\"\u003e\n\u003cli\u003eIf the commands you write in the text editor are in accordance with the commands above then after you compile them an image will definitely be formed like the first image above. The result is the same as the following image.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcHXhUYJVE84Z5Cyzwl9MSxUbKtoFFBTP2vSUH_-uIyCyIBOFN02N4ysagIQIlukLlDb79WvAdb4cZzyz8MjnJ5YnO0zFkIM7Ip6EAshsA-aPZQgsV7E3h4gZJkzya3PQZlelDfqM9-p8/w320-h99/Screenshot+2021-06-09+11.19.34.png' alt=\"gambar yin dan yang di latex\" \u003e\u003c/p\u003e\n\u003cp\u003eThe image of Yin and Yang above is in a vertical position, what if the image were in a horizontal position? Look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5fRKnUayzvZF1PsjCcWPzYfYJ_JVQlQf6BamQqmhQEUp1GNifu5PysPqPDcSWVZZKEYm5YTn6pdzNYcphROU_TV67LZNhUq3N0DFwjzbPd-oeLkRt1_hi303cbHRACtBP1TvHaSoSNXA/w320-h283/Screenshot+2021-06-09+11.34.03.png' alt=\"gambar yin dan yang\" \u003e\u003c/p\u003e\n\u003cp\u003eThe steps are as follows.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen your text editor.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eCreate a new project.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eAs usual, to draw, you need a TikZ package and include it in the preamble.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eNext, in the body of the document, the first image we create is a circle using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[thick]\u003c/span\u003e (0,0) circle (3)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eThen we will draw the head, this head is an arc starting from coordinates (-3, 0) via the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e(-3,0) arc (180:360:1.5) arc (180:0:1.5)\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eThen to draw the eyes, we will create two small circles using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e(1.5,0) circle (.5) (-1.5,0) circle (.5);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe results of the steps above look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUQgXCTljNBz5IVUFaknMSNThuPenxLKWq0bxh2gij93BvZJ_u-isIOA3aygQZIITClfMMO3fRBSJWMZf6zAQ_pftwPWdoW7CaMFdMpmT1wym-DJ_DxPO1FnFSgJHl81TvBpyRMcIF_N0/w320-h108/Screenshot+2021-06-09+13.28.25.png' alt=\"gambar yin dan yang di latex\" \u003e\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eNext, we will color the image that has been created. Here we will color the head and eyes in one part black using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[even odd rule]\u003c/span\u003e (1.5,0) circle (.5) (-3,0) arc (180:360:3) arc (0:180:1.5) arc (360:180:1.5) (-1.5,0) circle (.5);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eOf course you can give it a different color from the original color, you can use red, blue, yellow, green or other colors. The way to do this is to write the color you want before the text \u0026ldquo;even odd rule\u0026rdquo;, for example the color chosen is blue, then the text will be \\fill[blue, even odd rule] \u0026hellip;\u003c/p\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eOnce you have finished with all the steps above, it\u0026rsquo;s time to compile the script you created and see the results. Is it the same as the following picture.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEotgVFYbq7XstbdjrojSp-ZDDw9f8b3_JlcShVmneVfXBqi3eSrAM657mfdvmOLhGTAzflxZylb3lOOKHo0cpR3hmVEuAaNfAQOCiSk5vrKF6y5A5cR6k9JyUDkLyikXL3tI7oGXN9Wo/w320-h117/Screenshot+2021-06-09+13.37.07.png' alt=\"gambar yin dan yang di latex\" \u003e\u003c/p\u003e\n\u003cp\u003eSo we have applied it step by step in drawing the Yin and Yang symbols using LaTeX. If there are errors, please correct them, if you have input, please write them here. I hope this is helpful. Reference: \u003ca href=\"http://www.latexdraw.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ewww.latexdraw.com\u003c/a\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Two Tangent Sectors", "date_published" : "2023-03-01T13:42:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/dua-juring-bersinggungan/", "url" : "https://www.aantriono.com/en/dua-juring-bersinggungan/", "summary": " Drawing Two Tangent SectorsThis article shows how to draw two tangent sectors in LaTeX using TikZ. The figure comes from a geometry problem about the area between two sectors inside a square, but the goal here is only to reproduce the diagram cleanly.\nAfter testing the construction in Overleaf, the coordinates can be arranged precisely enough to make the figure look symmetric.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-two-tangent-sectors\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-two-tangent-sectors\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Two Tangent Sectors\"\u003e\u003c/a\u003eDrawing Two Tangent Sectors\u003c/h2\u003e\u003cp\u003eThis article shows how to draw two tangent sectors in LaTeX using TikZ. The figure comes from a geometry problem about the area between two sectors inside a square, but the goal here is only to reproduce the diagram cleanly.\u003c/p\u003e\n\u003cp\u003eAfter testing the construction in Overleaf, the coordinates can be arranged precisely enough to make the figure look symmetric.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPv8PHSxPKsjwHs2rhBlxt1hOoEeFay2UCCzeRpxZ0_Qbh9bEaz5Yd-V9SwlPsK5xOzgYSAv084R3p1BxrRCa-bNkQaoPAyxDJrURiesQ8_TqmUkBRP-bNA2lnp01VIiexa-7WWjuDKTM/s503/duabusur1.png' alt=\"Two tangent sectors in LaTeX\" \u003e\u003c/p\u003e\n\u003c!--more--\u003e\n\u003ch2 id=\"step-by-step-tikz-construction\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-tikz-construction\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step TikZ Construction\"\u003e\u003c/a\u003eStep-by-Step TikZ Construction\u003c/h2\u003e\u003cp\u003eFollow the steps below to recreate the figure.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen your LaTeX editor.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eLoad the \u003ccode\u003etikz\u003c/code\u003e and \u003ccode\u003etcolorbox\u003c/code\u003e packages in the preamble.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[many]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eDefine the \u003ccode\u003efrogbox\u003c/code\u003e style for the problem box.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\tcbset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e frogbox/.style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e enhanced,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colback=white!25!,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e colframe=green!80!black,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e enlarge top by=5.5mm,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e overlay=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e2cm,3.5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[shift=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e([xshift=\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e]frame.north west)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[draw=green!65!black,fill=gray!20,line width=1mm]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (0,0) arc (0:180:5mm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eAdd the small black eyes to complete the decoration.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\path\u003c/span\u003e\u003cspan class=\"na\"\u003e[fill=black]\u003c/span\u003e (-0.2,0) arc (0:180:1.2mm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}}}\u003c/span\u003e]\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eIn the document body, start the question box with the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etcolorbox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[frogbox,title=Calculate the area of the white region inside the red square.]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eNow define the coordinate points and their labels.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=left:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{A}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (A) at (0, 0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=right:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{B}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (B) at (4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5, 0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=right:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{C}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (C) at (4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5, 4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=left:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{D}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (D) at (0, 4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=left:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{E}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (E) at (0,4.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=above:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{F}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (F) at (4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5-4.5, 4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=right:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{G}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (G) at (4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5, 4.5*2\u003cspan class=\"nb\"\u003e^\u003c/span\u003e0.5-4.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\coordinate\u003c/span\u003e [label=below:\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"nv\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{H}\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e] (H) at (4.5,0);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAt this stage the sketch will look like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjqEII0lwFQX3CmnemUu7M8wBGzFTCMgADsPK5-cnyFQQppEilUYWRIqcBY231F10wi7D2CRtj2A2jhdAx6wYbfzqZB9WdMTfpanNYEApOWk-8ClaPgSQgkiplPRnG_VkMy0sW8ybXpms/s1166/Screenshot+2021-06-11+16.58.24.png' alt=\"Coordinate setup for the two sectors\" \u003e\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eDraw the first sector, \u003ccode\u003eAEH\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[black,fill=blue!75]\u003c/span\u003e (A)--(E) node[above,pos=0.5] \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e arc (90:0:4.5) -- cycle;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result will look like this.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi87Dlaj84ytRQVtQqR9knb8cf1w9jiv_2JTl2ErzhVU47zGoViYQG3U0c1HVfDEMg-axlc9bokl8Q90VHB2b7ZGysszqS_MymFnoBxahRxQSPkvbHjee1ko70s6w3MulDyJORrbieOGV8/s1166/Screenshot+2021-06-11+17.03.46.png' alt=\"First sector\" \u003e\u003c/p\u003e\n\u003cp\u003eThen draw the second sector, \u003ccode\u003eCFG\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\filldraw\u003c/span\u003e\u003cspan class=\"na\"\u003e[black,fill=blue!75]\u003c/span\u003e (C)--(G) node[above,pos=0.5] \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e arc (270:180:4.5) -- cycle;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eAdd the dimension arrow for \u003ccode\u003eAH\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[|{stealth}-{stealth}|]\u003c/span\u003e([yshift=-.6cm]A)--([yshift=-.6cm]H)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e node[below,midway,sloped]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e10\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"k\"\u003e\\,\u003c/span\u003ecm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAdd a second dimension arrow for \u003ccode\u003eCG\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e\u003cspan class=\"na\"\u003e[|{latex}-{latex}|]\u003c/span\u003e([xshift=.6cm]C)--([xshift=.6cm]G)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e node[above,midway,sloped]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"m\"\u003e10\u003c/span\u003e\u003cspan class=\"s\"\u003e$\u003c/span\u003e\u003cspan class=\"k\"\u003e\\,\u003c/span\u003ecm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eWith those steps, the diagram of two tangent sectors is complete. You can still adjust the colors, dimensions, and labels to match your own worksheet or teaching material.\u003c/p\u003e", "language": "en" }, { "title" : "Bar Charts", "date_published" : "2023-03-01T13:41:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/06/diagram-batang/", "url" : "https://www.aantriono.com/2021/06/diagram-batang/", "summary": " IntroductionThis article explains how to draw bar charts in LaTeX. The two most common forms are vertical bar charts and horizontal bar charts.\nVertical Bar ChartOne practical package for vertical bar charts is pgfplots.\nUse the article document class. latex\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThis article explains how to draw bar charts in LaTeX. The two most common forms are vertical bar charts and horizontal bar charts.\u003c/p\u003e\n\u003ch2 id=\"vertical-bar-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#vertical-bar-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Vertical Bar Chart\"\u003e\u003c/a\u003eVertical Bar Chart\u003c/h2\u003e\u003cp\u003eOne practical package for vertical bar charts is \u003ccode\u003epgfplots\u003c/code\u003e.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eUse the \u003ccode\u003earticle\u003c/code\u003e document class.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eAdd \u003ccode\u003epgfplots\u003c/code\u003e in the preamble.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfplots\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pgfplotsset\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ewidth=10cm,compat=1.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eSet up the axes and chart options.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eaxis\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ybar,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e enlargelimits=0.20,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ylabel=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAverage score\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e xlabel=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eStudent name\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e symbolic x coords=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAbdi,Andi,Bayu,Cindi,Clara,Diva\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e xtick=data,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e nodes near coords,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e nodes near coords align=\u003cspan class=\"nb\"\u003e{\u003c/span\u003evertical\u003cspan class=\"nb\"\u003e}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eAdd the data points.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\addplot\u003c/span\u003e coordinates \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (Abdi,65) (Andi,85) (Bayu,70)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e (Cindi,80) (Clara,60) (Diva,90)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eaxis\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eThe final result looks like this.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7EIfbgYCPnuY8GX7h7-dPQwwPX1m0M2Ts_-9Pzl4Sdq4fQr7pgcKF0Y3vJ5SnqDvNxflUEHYSCydep_uOdya5Jj14zJhY_orVnpU5vuR5zvceCBJnPvzgyC8fZhYLXz3us8mbtJxSeLE/s545/Screenshot+2021-06-12+10.59.55.png' alt=\"Vertical bar chart in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"horizontal-bar-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#horizontal-bar-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Horizontal Bar Chart\"\u003e\u003c/a\u003eHorizontal Bar Chart\u003c/h2\u003e\u003cp\u003eFor a horizontal bar chart, \u003ccode\u003etikz\u003c/code\u003e works well.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eKeep using the \u003ccode\u003earticle\u003c/code\u003e document class.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eAdd the required packages.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[indonesian]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eDefine colors for the bars.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eabdi\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eD7191C\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eandi\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eFDAE61\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebayu\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eABDDA4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecindi\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e2B83BA\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eclara\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e00FFFF\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ediva\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e8A2BE2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003ePrepare the \u003ccode\u003efigure\u003c/code\u003e environment.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eCreate the drawing scale with \u003ccode\u003etikzpicture\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[x=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(.1,0)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e ...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eDraw the \u003ccode\u003ex\u003c/code\u003e and \u003ccode\u003ey\u003c/code\u003e axes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) -- (100,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) -- (0,6.4);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eAdd scale labels on the \u003ccode\u003ex\u003c/code\u003e axis.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e10,20,...,100\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,.2) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,0) node[below] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eAdd the labels and bars on the \u003ccode\u003ey\u003c/code\u003e axis.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e\u003cspan class=\"na\"\u003e[count=\\y]\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Abdi/65/abdi,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Andi/85/andi,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Bayu/70/bayu,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Cindi/80/cindi,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Clara/60/clara,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Diva/90/diva\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[left]\u003c/span\u003e at (0,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\c]\u003c/span\u003e (0,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e-.4) rectangle (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e+.4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[right]\u003c/span\u003e at (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"9\"\u003e\n\u003cli\u003eAdd the caption.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAverage Mathematics Test Score Bar Chart\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"10\"\u003e\n\u003cli\u003eThe final result looks like this.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0_Vwl6dI6xMABjwHn2MPYNpAwgezE0ViCDtRWO3tDOu2wutZ5pTcdFVXoDPncmnAO-okC5Bnzkj7TkNOFNDOPf-_XDLXKzoyxCGmJbb6TkT1hmCzgOJfPuQES12fXRnqN1I5FliW8u6M/s680/Screenshot+2021-06-12+21.21.44.png' alt=\"Horizontal bar chart in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eHere is a complete example for the horizontal version.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[indonesian]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eabdi\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eD7191C\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eandi\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eFDAE61\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebayu\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eHTML\u003cspan class=\"nb\"\u003e}{\u003c/span\u003eABDDA4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan 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class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\centering\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[x=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e(.1,0)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) -- (100,0);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0,0) -- (0,6.4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\c\u003c/span\u003e\u003cspan class=\"na\"\u003e[count=\\y]\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Abdi/65/abdi,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Andi/85/andi,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Bayu/70/bayu,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Cindi/80/cindi,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Clara/60/clara,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Diva/90/diva\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[left]\u003c/span\u003e at (0,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[\\c]\u003c/span\u003e (0,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e-.4) rectangle (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e+.4);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[right]\u003c/span\u003e at (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\y\u003c/span\u003e) \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\x\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e10,20,...,100\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,.2) -- (\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e,0) node[below] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\x\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\caption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAverage Mathematics Test Score Bar Chart\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efigure\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThese two examples provide a solid starting point for making bar charts in LaTeX. From here, you can extend the style, colors, axis options, and labels as needed.\u003c/p\u003e\n", "language": "en" }, { "title" : "Pie Charts", "date_published" : "2023-03-01T13:40:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/06/diagram-lingkaran/", "url" : "https://www.aantriono.com/2021/06/diagram-lingkaran/", "summary": "This article continues the earlier LaTeX chart tutorials by focusing on pie charts and several related chart styles using the pgf-pie package.\nPie charts are commonly used to display proportions or comparisons. With pgf-pie, you can also create polar charts, cloud charts, and square charts from a similar syntax.\nChart TypesThis article covers:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article continues the earlier LaTeX chart tutorials by focusing on \u003cstrong\u003epie charts\u003c/strong\u003e and several related chart styles using the \u003ccode\u003epgf-pie\u003c/code\u003e package.\u003c/p\u003e\n\u003cp\u003ePie charts are commonly used to display proportions or comparisons. With \u003ccode\u003epgf-pie\u003c/code\u003e, you can also create polar charts, cloud charts, and square charts from a similar syntax.\u003c/p\u003e\n\u003ch2 id=\"chart-types\" class=\"headerLink\"\u003e\n \u003ca href=\"#chart-types\" class=\"header-mark\" aria-label=\"Direct link to section: Chart Types\"\u003e\u003c/a\u003eChart Types\u003c/h2\u003e\u003cp\u003eThis article covers:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003estandard pie charts\u003c/li\u003e\n\u003cli\u003epolar charts (\u003ccode\u003epolar area charts\u003c/code\u003e)\u003c/li\u003e\n\u003cli\u003ecloud charts\u003c/li\u003e\n\u003cli\u003esquare charts\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"standard-pie-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#standard-pie-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Standard Pie Chart\"\u003e\u003c/a\u003eStandard Pie Chart\u003c/h2\u003e\u003cp\u003eBasic steps:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eopen a LaTeX editor such as Overleaf\u003c/li\u003e\n\u003cli\u003euse the \u003ccode\u003earticle\u003c/code\u003e document class\u003c/li\u003e\n\u003cli\u003eload the \u003ccode\u003epgf-pie\u003c/code\u003e package\u003c/li\u003e\n\u003cli\u003edraw the chart inside a \u003ccode\u003etikzpicture\u003c/code\u003e environment\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eBasic example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgf-pie\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pie\u003c/span\u003e\u003cspan class=\"na\"\u003e[rotate=90]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e25/GeoGebra, 20/Blogger, 15/Edmodo, 20/\u003cspan class=\"k\"\u003e\\LaTeX\u003c/span\u003e, 20/Moodle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiscjIcIQftGEtRmH2nNuybyJXyT7qgETk0YNKTTlyZb1uxmmnppVGLD4ZGsdLkeN95zlnCulaTpCfi7DVz4aScInatuhyVY64c8DoqrKOGdoToX_RL07kRvp-NNB4z_4KsaEk4mw1a5p0/s455/Screenshot+2021-06-13+12.04.24.png' alt=\"Pie chart in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eIn this example, \u003ccode\u003erotate=90\u003c/code\u003e changes the starting position of the chart by \u003ccode\u003e90°\u003c/code\u003e.\u003c/p\u003e\n\u003ch2 id=\"polar-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#polar-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Polar Chart\"\u003e\u003c/a\u003ePolar Chart\u003c/h2\u003e\u003cp\u003eIn a polar chart, each sector has an angle and radius based on the represented value.\u003c/p\u003e\n\u003cp\u003eUse:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pie\u003c/span\u003e\u003cspan class=\"na\"\u003e[polar, explode=0.1, text=legend]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e25/GeoGebra, 20/Blogger, 15/Edmodo, 20/\u003cspan class=\"k\"\u003e\\LaTeX\u003c/span\u003e, 20/Moodle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eCompiled result:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNEyZOp-TU30ACJknDrFKqW1arBn2PHcnmyOsb6mdDFfdwHHBDJt0ibSe2MKpnh4xQrF7ynmOrEeeP0Wqv3TwnX8CuiOmWeXOcUg5Z7a0Qz-eLYCiKTIaGlBDBMTx9YiPWiAmPpAameI4/s462/Screenshot+2021-06-13+13.01.52.png' alt=\"Polar chart in LaTeX\" \u003e\u003c/p\u003e\n\u003cp\u003eIf you want the sectors to touch each other, remove \u003ccode\u003eexplode\u003c/code\u003e. If you want the labels inside the sectors, use \u003ccode\u003etext=inside\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pie\u003c/span\u003e[\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e polar,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e rotate=270,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e text=inside,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e color=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue!55, green!60, red!70, yellow!50, magenta!60\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e]\u003cspan class=\"nb\"\u003e{\u003c/span\u003e25/GeoGebra, 20/Blogger, 15/Edmodo, 20/\u003cspan class=\"k\"\u003e\\LaTeX\u003c/span\u003e, 20/Moodle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDWpTq8E-kac9pBI79IxIefTDDsxORC7ODMiD4jGx0l4pt50m2ilkam0ufNUYrBgFt8cbb1_chnH7olXlBxANg63fdOaj0tvq4sz0VgLmLCsiY63XuRi_dDOewV9Y4N2Q7HpI9dxGeB_c/s408/Screenshot+2021-06-13+21.02.16.png' alt=\"Colored polar chart in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"cloud-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#cloud-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Cloud Chart\"\u003e\u003c/a\u003eCloud Chart\u003c/h2\u003e\u003cp\u003eA cloud chart draws a set of disks whose sizes depend on the data values, with labels placed inside them.\u003c/p\u003e\n\u003cp\u003eCode:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pie\u003c/span\u003e\u003cspan class=\"na\"\u003e[cloud, text=inside, scale font, radius=3]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e25/GeoGebra, 20/Blogger, 15/Edmodo, 20/\u003cspan class=\"k\"\u003e\\LaTeX\u003c/span\u003e, 20/Moodle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUPBNH4NbMQ7HF-yCom4rREzT4RT6kNCLRr1hVMzoFIAtV8rixDRKtE-vvj-T4uVe-gQf9bJN7RMibqm7Q6faClhMzkUVLW9uPExlsLaMmqsxu7eb3eZ1egSFWm0dDcvjeK_XiT0KaOTc/s459/Screenshot+2021-06-13+13.13.29.png' alt=\"Cloud chart in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"square-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#square-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Square Chart\"\u003e\u003c/a\u003eSquare Chart\u003c/h2\u003e\u003cp\u003eThe package can also generate a square-style chart.\u003c/p\u003e\n\u003cp\u003eCode:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\pie\u003c/span\u003e\u003cspan class=\"na\"\u003e[square, scale font, color={blue!65, red!65, green, yellow, cyan}]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e25/GG, 20/Blogger, 15/Edmo, 20/\u003cspan class=\"k\"\u003e\\LaTeX\u003c/span\u003e, 20/Moodle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eCompiled result:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN_CybWeOQycRBIjHBOI7K4-4O-gWlNf22n5F6KvH2xgVM5Dvkj3JI9JICb9wCFJp2ilBYYFAa_IbFvanc6-2qDKaYkWF3quYc-p8cOfWZWV9fdo-x3PIcaYY4NlzTZ567bWCg35cpwUs/s424/Screenshot+2021-06-13+13.26.02.png' alt=\"Square chart in LaTeX\" \u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eWith \u003ccode\u003epgf-pie\u003c/code\u003e, creating statistical diagrams in LaTeX is fairly concise. You can adjust the rotation, colors, label positions, and chart type depending on the presentation you need.\u003c/p\u003e\n", "language": "en" }, { "title" : "Smart Diagrams", "date_published" : "2023-03-01T13:40:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/smart-diagram/", "url" : "https://www.aantriono.com/en/smart-diagram/", "summary": "This tutorial introduces the smartdiagram package in LaTeX. The package provides several ready-to-use diagram styles, which makes it useful when you want to add simple visual structure to a document without building everything manually in TikZ.\nTypes of Smart DiagramsSome of the available styles are:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis tutorial introduces the \u003ccode\u003esmartdiagram\u003c/code\u003e package in LaTeX. The package provides several ready-to-use diagram styles, which makes it useful when you want to add simple visual structure to a document without building everything manually in TikZ.\u003c/p\u003e\n\u003ch2 id=\"types-of-smart-diagrams\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-smart-diagrams\" class=\"header-mark\" aria-label=\"Direct link to section: Types of Smart Diagrams\"\u003e\u003c/a\u003eTypes of Smart Diagrams\u003c/h2\u003e\u003cp\u003eSome of the available styles are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eFlow diagram\u003c/li\u003e\n\u003cli\u003eCircular diagram\u003c/li\u003e\n\u003cli\u003eBubble diagram\u003c/li\u003e\n\u003cli\u003eDescriptive diagram\u003c/li\u003e\n\u003cli\u003eConstellation diagram\u003c/li\u003e\n\u003cli\u003ePriority descriptive diagram\u003c/li\u003e\n\u003cli\u003eAnimating diagram\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eBefore trying the examples below, prepare a basic document like this:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esmartdiagram\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e...\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"a-flow-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-flow-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: A. Flow Diagram\"\u003e\u003c/a\u003eA. Flow Diagram\u003c/h3\u003e\u003cp\u003eA flow diagram can be displayed horizontally or vertically.\u003c/p\u003e\n\u003cp\u003eHorizontal example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[flow diagram:horizontal]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Pedagogic, Personality, Professional, Social\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe output looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgToAYDoYA9fmbEdmMPJjvhUyBrwxlcyKn5teYuEB-NNZdxpdtW6yVvAT8y3gWexBlv1yf7HWo_MqgWlMEJhrlUgHMYJGtAJeMJ40s49OtgdOlIyQGSSfu0xLD1T9mPpv_wy6YWnOqiZfk/s797/Screenshot+2021-06-14+13.16.36.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"flow diagram in latex\" border=\"0\" data-original-height=\"200\" data-original-width=\"797\" height=\"100\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgToAYDoYA9fmbEdmMPJjvhUyBrwxlcyKn5teYuEB-NNZdxpdtW6yVvAT8y3gWexBlv1yf7HWo_MqgWlMEJhrlUgHMYJGtAJeMJ40s49OtgdOlIyQGSSfu0xLD1T9mPpv_wy6YWnOqiZfk/w400-h100/Screenshot+2021-06-14+13.16.36.png\" title=\"flow diagram\" width=\"400\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eTo switch to a vertical layout, replace \u003ccode\u003ehorizontal\u003c/code\u003e with \u003ccode\u003evertical\u003c/code\u003e.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[flow diagram:vertical]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Pedagogic, Personality, Professional, Social\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoyhMKpq6nbvKdERbNQIgQhyphenhyphenDnRPKg562QIf6hteEiqvhEUDz0qELPS-_muhJpcYCKtJ2topuq3YdWBKb86oQgpNdJbPgaXiQ4NEZIBW7Vd806HMDm8JiMnsm4gos33JR9FkbAjqiwvB8/s457/Screenshot+2021-06-14+13.21.43.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"vertical flow diagram in latex\" border=\"0\" data-original-height=\"457\" data-original-width=\"271\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoyhMKpq6nbvKdERbNQIgQhyphenhyphenDnRPKg562QIf6hteEiqvhEUDz0qELPS-_muhJpcYCKtJ2topuq3YdWBKb86oQgpNdJbPgaXiQ4NEZIBW7Vd806HMDm8JiMnsm4gos33JR9FkbAjqiwvB8/w190-h320/Screenshot+2021-06-14+13.21.43.png\" title=\"vertical flow diagram\" width=\"190\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"b-circular-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-circular-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: B. Circular Diagram\"\u003e\u003c/a\u003eB. Circular Diagram\u003c/h3\u003e\u003cp\u003eA circular diagram places items around a circle.\u003c/p\u003e\n\u003cp\u003eClockwise rotation:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[circular diagram:clockwise]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Pedagogic, Personality, Professional, Social\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijyIsvPZ-eBq9RF4an9JpNL3DBosVSKIGH-_K-S5QG7086bCtmsW7hIy3Wl8kVxkcfRFyAjGK8gvA-WduswDaDV0cUh8Qmfo37GAAMUDZAryJnTNyeCL40uRq_DrDJF8kIYtCdA77MSbI/s473/Screenshot+2021-06-14+13.26.38.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"circular diagram in latex\" border=\"0\" data-original-height=\"456\" data-original-width=\"473\" height=\"308\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijyIsvPZ-eBq9RF4an9JpNL3DBosVSKIGH-_K-S5QG7086bCtmsW7hIy3Wl8kVxkcfRFyAjGK8gvA-WduswDaDV0cUh8Qmfo37GAAMUDZAryJnTNyeCL40uRq_DrDJF8kIYtCdA77MSbI/w320-h308/Screenshot+2021-06-14+13.26.38.png\" title=\"circular diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eCounterclockwise rotation:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[circular diagram:anticlockwise]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Pedagogic, Personality, Professional, Social\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-l0xTVI6bp7-dHh2f0kYJC89fWHHquRbR3_ptDGDwRZp4NJQU1EaBungtnBxDD7Jbq9E_ZaEInlGM81J0MCS99ogexIaVSShKqjzRr6cYNYqENq3mQJdch1QJhIkaeeuylBNumK_Sjtw/s473/Screenshot+2021-06-14+13.29.17.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"circular diagram in latex\" border=\"0\" data-original-height=\"456\" data-original-width=\"473\" height=\"308\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-l0xTVI6bp7-dHh2f0kYJC89fWHHquRbR3_ptDGDwRZp4NJQU1EaBungtnBxDD7Jbq9E_ZaEInlGM81J0MCS99ogexIaVSShKqjzRr6cYNYqENq3mQJdch1QJhIkaeeuylBNumK_Sjtw/w320-h308/Screenshot+2021-06-14+13.29.17.png\" title=\"circular diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"c-bubble-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-bubble-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: C. Bubble Diagram\"\u003e\u003c/a\u003eC. Bubble Diagram\u003c/h3\u003e\u003cp\u003eA bubble diagram places one main concept in the center and supporting items around it.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[bubble diagram]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Teacher Competence, Pedagogic, Personality, Professional, Social\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb38ITaPUvrTGvz5poa4pNxq3-44rZLLIxl5gCDbez11JTjIIY-qymfil_a4MmsDG8190j73XUwgYQg7nfGfSqP4y-Bb6rU4v7H0Dl0ttXB8aWI8hNLl-JSgHJtogZ_eybjte4akS8834/s560/Screenshot+2021-06-14+13.33.57.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bubble diagram in latex\" border=\"0\" data-original-height=\"541\" data-original-width=\"560\" height=\"309\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjb38ITaPUvrTGvz5poa4pNxq3-44rZLLIxl5gCDbez11JTjIIY-qymfil_a4MmsDG8190j73XUwgYQg7nfGfSqP4y-Bb6rU4v7H0Dl0ttXB8aWI8hNLl-JSgHJtogZ_eybjte4akS8834/w320-h309/Screenshot+2021-06-14+13.33.57.png\" title=\"bubble diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"d-descriptive-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-descriptive-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: D. Descriptive Diagram\"\u003e\u003c/a\u003eD. Descriptive Diagram\u003c/h3\u003e\u003cp\u003eA descriptive diagram is useful when each item also needs a short explanation.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[descriptive diagram]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003ePedagogic,\u003cspan class=\"nb\"\u003e{\u003c/span\u003eThe teacher\u0026#39;s ability to manage the learning process\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003ePersonality,\u003cspan class=\"nb\"\u003e{\u003c/span\u003ePersonal character that reflects a good attitude\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003eProfessional,\u003cspan class=\"nb\"\u003e{\u003c/span\u003eDeep and broad mastery of the subject matter\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003eSocial,\u003cspan class=\"nb\"\u003e{\u003c/span\u003eThe ability to communicate and interact effectively\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiczFPp_EDWsMoGz6kjrIfyE0klX70Ccp10eyIb0QHNDjDyUWLAYnCkZbnM35Z93QI0I9MlGupQO7F4ai4yFNrLaMgtILR-u_gszE6pIrK0IpRSgo1-NHQIGzMeNRWgFP9l4EVq8t3Hstw/s510/Screenshot+2021-06-14+14.23.57.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"descriptive diagram in latex\" border=\"0\" data-original-height=\"487\" data-original-width=\"510\" height=\"306\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiczFPp_EDWsMoGz6kjrIfyE0klX70Ccp10eyIb0QHNDjDyUWLAYnCkZbnM35Z93QI0I9MlGupQO7F4ai4yFNrLaMgtILR-u_gszE6pIrK0IpRSgo1-NHQIGzMeNRWgFP9l4EVq8t3Hstw/w320-h306/Screenshot+2021-06-14+14.23.57.png\" title=\"descriptive diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"e-constellation-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#e-constellation-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: E. Constellation Diagram\"\u003e\u003c/a\u003eE. Constellation Diagram\u003c/h3\u003e\u003cp\u003eA constellation diagram places one core item in the center and connects surrounding items with arrows.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[constellation diagram]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Teacher Competence, Pedagogic, Personality, Professional, Social\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZtChbbcmMpDqhB-XTYiB6W7ZUC2JDjFTPZvShkeAoTJ480bz7eTrlAXskVDItBiG_2Wff8M3KtS_eS2FCMiIoK8-yIb-HBALROxs-wrMbkFncz3ZbGypqrZ6OVm33Bt5FnU8faXCH9ik/s560/Screenshot+2021-06-14+13.37.53.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"constellation diagram in latex\" border=\"0\" data-original-height=\"557\" data-original-width=\"560\" height=\"318\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZtChbbcmMpDqhB-XTYiB6W7ZUC2JDjFTPZvShkeAoTJ480bz7eTrlAXskVDItBiG_2Wff8M3KtS_eS2FCMiIoK8-yIb-HBALROxs-wrMbkFncz3ZbGypqrZ6OVm33Bt5FnU8faXCH9ik/w320-h318/Screenshot+2021-06-14+13.37.53.png\" title=\"constellation diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"f-priority-descriptive-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#f-priority-descriptive-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: F. Priority Descriptive Diagram\"\u003e\u003c/a\u003eF. Priority Descriptive Diagram\u003c/h3\u003e\u003cp\u003eThis diagram style is useful when you want to emphasize a sequence or hierarchy.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagram\u003c/span\u003e\u003cspan class=\"na\"\u003e[priority descriptive diagram]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Planning instruction,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Delivering instruction,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Evaluating instruction,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Assessing outcomes,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Analyzing outcomes,\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Following up the outcomes\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcOWqAAXfEOIrJXy1kHN0S_zTwClj8WaoYifFRAHaLZStFo9_-_QQh1CufECEOE_GESzdUa1YP3qRD3M6ulophsHNi-lY5J5Ur5QAL66iql0rEX0Fg_yorxnRKrqxX0meqUeiXVQ2SPZs/s570/Screenshot+2021-06-14+14.36.35.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"priority descriptive diagram in latex\" border=\"0\" data-original-height=\"570\" data-original-width=\"400\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcOWqAAXfEOIrJXy1kHN0S_zTwClj8WaoYifFRAHaLZStFo9_-_QQh1CufECEOE_GESzdUa1YP3qRD3M6ulophsHNi-lY5J5Ur5QAL66iql0rEX0Fg_yorxnRKrqxX0meqUeiXVQ2SPZs/w225-h320/Screenshot+2021-06-14+14.36.35.png\" title=\"priority descriptive diagram\" width=\"225\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"g-animating-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#g-animating-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: G. Animating Diagram\"\u003e\u003c/a\u003eG. Animating Diagram\u003c/h3\u003e\u003cp\u003eAn animating diagram is usually used in a \u003ccode\u003ebeamer\u003c/code\u003e presentation. The diagram appears step by step across slides.\u003c/p\u003e\n\u003cp\u003eBasic example:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebeamer\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003esmartdiagram\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eframe\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\smartdiagramanimated\u003c/span\u003e\u003cspan class=\"na\"\u003e[circular diagram]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e Pedagogic, Personality, Professional, Social\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eframe\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eExample output:\u003c/p\u003e\n\u003ctable\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0OkHe5VsqSpCz5k4nRZv4cVzaSsue4JjH5E-AV0C5Kg5Cza7SWDgqJIy095TLPXvlHsM8ueJ4wP4z9jj2nVJtH-NKJ_JPYk__51U6frtrurMRd4t_98iClK3COd-nZhS2_akvud5I9JM/s593/Screenshot+2021-06-14+14.44.33.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"animated diagram in latex\" border=\"0\" data-original-height=\"446\" data-original-width=\"593\" height=\"241\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0OkHe5VsqSpCz5k4nRZv4cVzaSsue4JjH5E-AV0C5Kg5Cza7SWDgqJIy095TLPXvlHsM8ueJ4wP4z9jj2nVJtH-NKJ_JPYk__51U6frtrurMRd4t_98iClK3COd-nZhS2_akvud5I9JM/w320-h241/Screenshot+2021-06-14+14.44.33.png\" title=\"animated diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/td\u003e\u003ctd\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTtm-Dhhuz0sh-yTE1HMh2tUhwtBo8gGukc8YjAL4hLDwTMNpJz9aQR07En8-zhbMug_N8RqRv5ysUMgzELoiMkVlgAzK-CGY7wvFkoNkIay1xe0gEg9thcBrd-2MmlwnsijKTWYSRElw/s586/Screenshot+2021-06-14+14.45.03.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"animated diagram in latex\" border=\"0\" data-original-height=\"446\" data-original-width=\"586\" height=\"244\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTtm-Dhhuz0sh-yTE1HMh2tUhwtBo8gGukc8YjAL4hLDwTMNpJz9aQR07En8-zhbMug_N8RqRv5ysUMgzELoiMkVlgAzK-CGY7wvFkoNkIay1xe0gEg9thcBrd-2MmlwnsijKTWYSRElw/w320-h244/Screenshot+2021-06-14+14.45.03.png\" title=\"animated diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/td\u003e\u003ctd\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgP0yS3ogHay47CLmHsWDJp9RQ1N7qJKV85-WBWY5msXmUJ5tOKTv70AozWyxrrLvY8rk6TxTV-wWGtay9PCmw_-4e94dmYVBg3E2jFWMlTGonRgLfWIHaA8ckxfmrlaZ3M0MU3lcJYjsc/s592/Screenshot+2021-06-14+14.45.48.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"animated diagram in latex\" border=\"0\" data-original-height=\"447\" data-original-width=\"592\" height=\"242\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgP0yS3ogHay47CLmHsWDJp9RQ1N7qJKV85-WBWY5msXmUJ5tOKTv70AozWyxrrLvY8rk6TxTV-wWGtay9PCmw_-4e94dmYVBg3E2jFWMlTGonRgLfWIHaA8ckxfmrlaZ3M0MU3lcJYjsc/w320-h242/Screenshot+2021-06-14+14.45.48.png\" title=\"animated diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/td\u003e\u003ctd\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0ov5R_PNdWMe_y5fPjlUzmOgFEtCqgkEStVIpU51K2oYcWo7o3OJYQZKs05-_q9z4mo7xH8cxkK4ggN3Tjut5dHPL-KWJY7QlA15a5TxnD0myff0mhCghqqJ4xP8xDGHVei_RyYdcBC4/s592/Screenshot+2021-06-14+14.46.03.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"animated diagram in latex\" border=\"0\" data-original-height=\"447\" data-original-width=\"592\" height=\"242\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0ov5R_PNdWMe_y5fPjlUzmOgFEtCqgkEStVIpU51K2oYcWo7o3OJYQZKs05-_q9z4mo7xH8cxkK4ggN3Tjut5dHPL-KWJY7QlA15a5TxnD0myff0mhCghqqJ4xP8xDGHVei_RyYdcBC4/w320-h242/Screenshot+2021-06-14+14.46.03.png\" title=\"animated diagram\" width=\"320\" /\u003e\u003c/a\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eThe \u003ccode\u003esmartdiagram\u003c/code\u003e package offers several ready-made layouts and is a practical choice when you need diagrams quickly inside a LaTeX document or presentation. The colors in these examples are the package defaults, but they can be customized further if needed.\u003c/p\u003e\n", "language": "en" }, { "title" : "Working With LaTeX Offline", "date_published" : "2023-03-01T13:40:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/07/bekerja-dengan-latex-secara-offline/", "url" : "https://www.aantriono.com/2021/07/bekerja-dengan-latex-secara-offline/", "summary": "This article explains how to work with LaTeX offline. For online use, Overleaf is a common option. To work offline, however, you need to install the necessary software on your own computer.\nIn general, an offline LaTeX setup has three main parts:\nLaTeX engine LaTeX editor LaTeX packages The LaTeX engine is the core program that receives commands, compiles the document, and produces the output. The LaTeX editor is the interface where users write the source text and LaTeX commands. LaTeX packages are collections of extra files that provide additional features.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eThis article explains how to work with LaTeX offline. For online use, Overleaf is a common option. To work offline, however, you need to install the necessary software on your own computer.\u003c/p\u003e\n\u003cp\u003eIn general, an offline LaTeX setup has three main parts:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eLaTeX engine\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLaTeX editor\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLaTeX packages\u003c/strong\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThe LaTeX engine is the core program that receives commands, compiles the document, and produces the output. The LaTeX editor is the interface where users write the source text and LaTeX commands. LaTeX packages are collections of extra files that provide additional features.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE8l0V6dma6rYrAYZDZ1oB2RWOGPK3z7XAP4fYLwW3OjrEU5_4wNDVAZ4VBHzSJ8EORvq9MisP22Q03IbHX33cbWoW-1ahEid1ORXsP37i6EBbm_aJoGgTdBuviqRcnTnPZalzIxsB8n0/s460/Screenshot+2021-07-02+23.02.28.png' alt=\"LaTeX logo\" \u003e\u003c/p\u003e\n\u003cp\u003eOne helpful analogy is:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe \u003cstrong\u003eLaTeX engine\u003c/strong\u003e is like a chef\u003c/li\u003e\n\u003cli\u003ethe \u003cstrong\u003eLaTeX editor\u003c/strong\u003e is like the order sheet\u003c/li\u003e\n\u003cli\u003ethe \u003cstrong\u003eLaTeX package\u003c/strong\u003e is like a recipe or menu book used when special instructions are needed\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIn many basic cases, the engine can handle standard commands without much extra help. But as soon as you need colors, diagrams, advanced tables, or specialized notation, packages become important.\u003c/p\u003e\n\u003ch2 id=\"latex-engine\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-engine\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Engine\"\u003e\u003c/a\u003eLaTeX Engine\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinMTx1lNGY872MVPb-64UXeo1BjF3roe0d5Js2FgeubZevXEkuIVbYhgdyIE3LEp2T04XJckR0MIkhiGU5I_eNCBZPtUAuhLxB44ZoqKLOws4txf3kMGcfEcLU8j-IY8BQ-OK9eNeXmak/s717/Screenshot+2021-07-02+22.50.42+%25281%2529.png' alt=\"LaTeX engine TeX Live\" \u003e\u003c/p\u003e\n\u003cp\u003eA LaTeX engine is the program that compiles LaTeX source into output documents, usually PDF files.\u003c/p\u003e\n\u003cp\u003eCommon distributions include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eMiKTeX\u003c/strong\u003e for Windows\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eMacTeX\u003c/strong\u003e for macOS\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTeX Live\u003c/strong\u003e for Linux and Unix-like systems\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThese distributions can be downloaded from their official websites. Many Linux users prefer \u003cstrong\u003eTeX Live\u003c/strong\u003e because it is widely available and includes a large collection of packages.\u003c/p\u003e\n\u003ch2 id=\"latex-editor\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-editor\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Editor\"\u003e\u003c/a\u003eLaTeX Editor\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiT4R2LFJXPvz0e_AYhcQlkpk3JXU7sovNSWjskyQRt6jOPmlsR7UOZxURb2Ex6f23F6WPdOAo1ei1GMLZmLcVJ1Eappg7SjvZgH7dTMNQXoE3ZpiHQM55sm3azzBzIZ9mErcV9PvFK9WE/s460/Screenshot+2021-07-02+22.57.12+%25281%2529.png' alt=\"TeX editor TeXstudio\" \u003e\u003c/p\u003e\n\u003cp\u003eA LaTeX editor is the place where users write LaTeX source code. Technically, almost any text editor can be used, even a simple one like Notepad. But editors designed for LaTeX are much more convenient because they often provide:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003esyntax highlighting\u003c/li\u003e\n\u003cli\u003ecompile buttons\u003c/li\u003e\n\u003cli\u003ecommand completion\u003c/li\u003e\n\u003cli\u003eerror navigation\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eAfter writing the source, the document is usually saved with the \u003ccode\u003e.tex\u003c/code\u003e extension. This \u003ccode\u003e.tex\u003c/code\u003e file is the LaTeX source file that gets sent to the engine for compilation.\u003c/p\u003e\n\u003cp\u003ePopular LaTeX editors include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eTeXstudio\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTeXmaker\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTeXworks\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eTeXnicCenter\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eLyX\u003c/strong\u003e for a more WYSIWYG-like workflow\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eTeXstudio is especially popular because it is free, feature-rich, and comfortable for daily use.\u003c/p\u003e\n\u003ch2 id=\"latex-packages\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-packages\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Packages\"\u003e\u003c/a\u003eLaTeX Packages\u003c/h2\u003e\u003cp\u003eLaTeX packages are collections of small files containing macros or extra commands. These files usually have the \u003ccode\u003e.sty\u003c/code\u003e extension.\u003c/p\u003e\n\u003cp\u003eFor example, if you want colored text, you need to load a package such as \u003ccode\u003ecolor\u003c/code\u003e or \u003ccode\u003excolor\u003c/code\u003e. Once the package is called in the document, the LaTeX engine loads the corresponding file and gains access to those extra commands.\u003c/p\u003e\n\u003cp\u003ePackages support a huge range of tasks, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ecoloring text\u003c/li\u003e\n\u003cli\u003ebuilding tables\u003c/li\u003e\n\u003cli\u003edrawing charts\u003c/li\u003e\n\u003cli\u003ewriting advanced mathematical notation\u003c/li\u003e\n\u003cli\u003etypesetting Arabic text\u003c/li\u003e\n\u003cli\u003ewriting musical notation\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eBecause there are so many packages, a full LaTeX distribution can be quite large. That is one reason offline LaTeX installations often require significant disk space.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eWorking with LaTeX offline means writing and compiling documents directly on your own machine, without relying on a web service. This setup is useful for people who want a self-contained workflow, faster local compilation, or the ability to work without an internet connection.\u003c/p\u003e\n\u003cp\u003eWhether you prefer an online or offline workflow depends on your needs. The important part is understanding the role of the engine, the editor, and the packages so that the overall LaTeX workflow makes sense.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eReference:\u003c/strong\u003e Koredianto Usman, \u003cem\u003eA Practical Introduction to Using LaTeX for Scientific Reports and Journals\u003c/em\u003e.\u003c/p\u003e\n", "language": "en" }, { "title" : "Installing LaTeX Engine on Blog", "date_published" : "2023-03-01T13:39:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/memasang-latex-engine-di-blog/", "url" : "https://www.aantriono.com/en/memasang-latex-engine-di-blog/", "summary": " Introduction\nOn this occasion, I will only tell a story, when he first started learning LaTeX at the end of last year, I looked for references on Google and found several blogs that had LaTeX Engine installed in them, this is certainly one of the attractions and makes it very easy for LaTeX students who want to know the results of writing their LaTeX code. For example, the Blog https://www.latex4technics.com, and https://www.latexdraw.com. Begins from Curiosity and curiosity about how to install a LaTeX \u0026rsquo;engine\u0026rsquo; or LaTeX Engine on a Blog like the one on this Blog, I searched for various online literature and asked fellow bloggers, but had not succeeded in finding a way. Then, armed with courage, on Friday, July 16 2021, I had the courage to write an electronic letter (email) to the Blog I www.latexdraw.com\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch4 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h4\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpW4jUpGzgW-LtbOsLVYgyKeE4xNrQKW2K1cNsZxXxYFfXW2WHrKzymY8EQeI7r1iVbTptIKTr9Oa-TN-1HpRD2_BpvSI8GVvHJMnbeUfoISDm2pL0S0Hj8IGH6nK_09hhslkRNEoNFmQ/s1280/WhatsApp+Image+2021-08-18+at+14.36.11.jpeg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"memasang mesin latex di blog\" border=\"0\" data-original-height=\"1280\" data-original-width=\"1280\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpW4jUpGzgW-LtbOsLVYgyKeE4xNrQKW2K1cNsZxXxYFfXW2WHrKzymY8EQeI7r1iVbTptIKTr9Oa-TN-1HpRD2_BpvSI8GVvHJMnbeUfoISDm2pL0S0Hj8IGH6nK_09hhslkRNEoNFmQ/w320-h320/WhatsApp+Image+2021-08-18+at+14.36.11.jpeg\" title=\"Mesin latex di blog\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eOn this occasion, I will only tell a story, when he first started learning LaTeX at the end of last year, I looked for references on Google and found several blogs that had LaTeX Engine installed in them, this is certainly one of the attractions and makes it very easy for LaTeX students who want to know the results of writing their LaTeX code. For example, the Blog \u003ca href=\"https://www.latex4technics.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.latex4technics.com\u003c/a\u003e, and \u003ca href=\"https://www.latexdraw.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.latexdraw.com\u003c/a\u003e. Begins from Curiosity and curiosity about how to install a LaTeX \u0026rsquo;engine\u0026rsquo; or LaTeX Engine on a Blog like the one on this Blog, I searched for various online literature and asked fellow bloggers, but had not succeeded in finding a way. Then, armed with courage, on Friday, July 16 2021, I had the courage to write an electronic letter (email) to the Blog I \u003ca href=\"https://www.latexdraw.com\" target=\"_blank\"\u003e\u003ca href=\"https://www.latexdraw.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ewww.latexdraw.com\u003c/a\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e, namely Mr. Muhammed Benmiloud. Unexpectedly, he replied to his email and provided a link to an article on Github. On Saturday, August 14 2021 I contacted a blogger (pen name \u003ca href=\"https://www.rumusstatistik.com/\" target=\"_blank\"\u003eR-Stats\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e) via the contact provided on his blog, asking how he installed the CodeCogs LaTeX Editor page on his blog. Hmmm, apparently he is a statistics lecturer in Yogyakarta. Unexpectedly, he provided a script to install CodeCogs LaTeX Editor on the blog. And successfully installed on this blog, I deliberately named it MaTeX, an acronym for Mathematics and TeX. To try writing it or just take a look, please check it out \u003ca href=\"https://www.silagebra.com/p/matex.html\" target=\"_blank\"\u003ehere.\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThen on Sunday, August 15 2021, I sent a link from Mr. Muhammed Benmiloud told him to learn, it didn\u0026rsquo;t take much time, he could find the script (lecturers are different) and try it on his blog and it worked 100%. Not waiting long, I was given the script and installed it on the blog \u003ca href=\"http://www.aantriono.com\" target=\"_blank\"\u003e\u003ca href=\"https://www.aantriono.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ewww.aantriono.com\u003c/a\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e. Saluuuut to him, really cool. 👍.\u003c/p\u003e\n\u003ch4 id=\"latex-engine\" class=\"headerLink\"\u003e\n \u003ca href=\"#latex-engine\" class=\"header-mark\" aria-label=\"Direct link to section: LaTeX Engine\"\u003e\u003c/a\u003eLaTeX Engine\u003c/h4\u003e\u003cp\u003eActually, what is LaTeX Engine or LaTeX engine? You can read the explanation \u003ca href=\"https://tex.stackexchange.com/questions/13593/the-differences-between-tex-engines\" target=\"_blank\"\u003ehere,\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eor \u003ca href=\"http://tug.org/levels.html\" target=\"_blank\"\u003ethere.\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eSo far, to use LaTeX online, I use the site \u003ca href=\"https://www.overleaf.com\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ewww.overleaf.com\u003c/a\u003e, and offline uses TeXLive. In the LaTeX engine there are various options available, whether to use pdfTex, XeLaTeX, LuaLaTeX or others. As a test material, please copy the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e Kode Bendera Turki ini telah I publish sebelumnya.\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efancybox\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[x11names,dvipsnames]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebulanbintang\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e255,255,255\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eshapes.geometric\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\title\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eBendera Turki\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\author\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eAan Triono \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\date\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\today\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\centering\\maketitle\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e [red] (-5,-4) rectangle (8.5,3.5);\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[bulanbintang,even odd rule]\u003c/span\u003e (0,0) circle (2.5);\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[red]\u003c/span\u003e (.54,0) circle (2);\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[bulanbintang]\u003c/span\u003e (2.9,0) node [ regular polygon, regular polygon sides=5, inner sep=0pt, minimum size=1.5cm, shape border rotate=20] (A) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e (A.corner 1)\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1, 3, 5, 2, 4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-- (A.corner \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e) \u003cspan class=\"nb\"\u003e}\u003c/span\u003e -- cycle;\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eNext, open the LaTeX Engine page on this blog and delete all the existing code and replace or paste all the code, then click \u003ckbd\u003eRun at This Site.\u003c/kbd\u003e See the output on the page below. If there are no typing errors, the result will be as shown in the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipPB8myY9MNWAXqMN4pMXqXMlR23e3wJo3UJFPWW5K8cxljxjSi7MOPdPqFzO-615xpTdLOqfswbcSWd1DKHY4aJ5lnShyMDdhMmJ9RfJLRfkKBz4aJdURPKDbzzj6DT-o4eHBH6UklzA/s862/uji2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Memasang mesin LaTeX di blog\" border=\"0\" data-original-height=\"528\" data-original-width=\"862\" height=\"196\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipPB8myY9MNWAXqMN4pMXqXMlR23e3wJo3UJFPWW5K8cxljxjSi7MOPdPqFzO-615xpTdLOqfswbcSWd1DKHY4aJ5lnShyMDdhMmJ9RfJLRfkKBz4aJdURPKDbzzj6DT-o4eHBH6UklzA/w320-h196/uji2.png\" title=\"LaTeX Engine di Blog\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eApart from that there are options \u003ckbd\u003eOpen in Overleaf\u003c/kbd\u003e, if we click on the text, the LaTeX code will automatically be saved in our overleaf account. Next, here is the LaTeX code that comes from his blog \u003ca href=\"https://www.texblog.net\" target=\"_blank\"\u003eMr Stefan Kottwitz.\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epgfplots\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepgfplotslibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epolar,colormaps\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epolaraxis\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[ domain = -14400:14400, samples = 3000, colormap/cool, hide axis ] \u003cspan class=\"k\"\u003e\\addplot\u003c/span\u003e\u003cspan class=\"na\"\u003e[no markers,mesh,opacity=0.5]\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1-sin(40*x/39\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003epolaraxis\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf there are no typing errors, the result will be as shown in the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFReSpAp6Up7ZLchGyRL1aLBTJrzBLJTs67jzQSe-n4_MjZHZo2DJQu5-cxQtrg1UkUKTn1wO4vMyeVlZBCwtNGbqJOtedap9sgrnu_VogeyGSWS8VPAOfxSi-AIQZEvymEh6I90aC3uc/s867/uji1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Memasang mesin laTeX di blog\" border=\"0\" data-original-height=\"528\" data-original-width=\"867\" height=\"195\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFReSpAp6Up7ZLchGyRL1aLBTJrzBLJTs67jzQSe-n4_MjZHZo2DJQu5-cxQtrg1UkUKTn1wO4vMyeVlZBCwtNGbqJOtedap9sgrnu_VogeyGSWS8VPAOfxSi-AIQZEvymEh6I90aC3uc/w320-h195/uji1.png\" title=\"Mesin laTeX di blog\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch4 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h4\u003e\u003cp\u003eAfter successfully installing LaTeX Engine on the blog, and seeing the results, we can conclude Indeed, the compilation results are the same as when writing the code yourself on the overleaf.com site or on TeXLive. It\u0026rsquo;s just that there are still shortcomings and I have also asked about this via electronic mail with the help of Mrs. Yeni Sutrisni to Mr. Muhammed Benmiloud, namely: How do I upload files into the LaTeX Engine on a blog? Do you know the solution? Please write it in the comments for us to discuss. That\u0026rsquo;s the article about installing a LaTeX Engine or Latex Engine on a blog using a script from GitHub. Thank you. I hope this is helpful. Thanks to : Mr. Muhammed Benmiloud, Mr. R-Stats Lecturer.\u003c/p\u003e\n", "language": "en" }, { "title" : "Circle Optical Illusion", "date_published" : "2023-03-01T13:38:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/ilusi-optik-lingkaran/", "url" : "https://www.aantriono.com/en/ilusi-optik-lingkaran/", "summary": " IntroductionWhen we see images in print or electronic media, we definitely have or often see images that deceive the eye or what is usually called optical illusion. Actually, an optical illusion image is just an ordinary image, what makes it appear as if it moves or changes color is the ability of our eyes. One of the illusory images is the circular optical illusion below.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eWhen we see images in print or electronic media, we definitely have or often see images that deceive the eye or what is usually called \u003cem\u003eoptical illusion\u003c/em\u003e. Actually, an optical illusion image is just an ordinary image, what makes it appear as if it moves or changes color is the ability of our eyes. One of the illusory images is the circular optical illusion below.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEgdybYSHE9R9FqDu_gdLi4mMcPUmkUFPUlRSSY7qtC7jV2rL2SzC_1NpLkCij-afZE4XccdE0PeDMHjkmb3-sPK6mwXPx89EgsWewv7nS5f5hgGUJAerrBFZ1qNA9OVs-jzKKmeRcon_OHmM1FdFEgl5ycMcA2vM84dXGntZ14JpMXzaMxhus_sS_Gx=w680-h382-p-k-no-nu\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"ilusi optik lingkaran di latex\" border=\"0\" data-original-height=\"280\" data-original-width=\"563\" height=\"199\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEgdybYSHE9R9FqDu_gdLi4mMcPUmkUFPUlRSSY7qtC7jV2rL2SzC_1NpLkCij-afZE4XccdE0PeDMHjkmb3-sPK6mwXPx89EgsWewv7nS5f5hgGUJAerrBFZ1qNA9OVs-jzKKmeRcon_OHmM1FdFEgl5ycMcA2vM84dXGntZ14JpMXzaMxhus_sS_Gx=w400-h199\" title=\"ilusi optik lingkaran\" width=\"400\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eHow to draw the circular optical illusion using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e? Check out the following guide.\u003c/p\u003e\n\u003cdiv class=\"bacajuga\"\u003e\n\u003ch4\u003eAlso Read\u003c/h4\u003e\n\u003cul\u003e \u003cli\u003e\u003ca href=\"/2021/06/diagram-lingkaran/\"\u003ePie Chart\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"/2021/06/dua-juring-bersinggungan/\"\u003eTwo Sides Intersect\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"/2021/04/cara-menggambar-unsur-unsur-lingkaran/\"\u003eDrawing Circle Elements\u003c/a\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/div\u003e\n\u003ch2 id=\"drawing-the-optical-illusion-of-a-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-the-optical-illusion-of-a-circle\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing the Optical Illusion of a Circle\"\u003e\u003c/a\u003eDrawing the Optical Illusion of a Circle\u003c/h2\u003e\u003cp\u003eFor the first part, in the preamble, we write the following commands.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eNext, in the body of the document, we will start giving commands to draw using the following commands.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[line width=1mm, blue]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIn this command, the width of the circle is 1 mm and the color of the entire circle drawn is blue. Of course, you can change these two things according to your wishes. Arriving at the core part, draw 10 circles side by side using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (2.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (3.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (4.2,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (5,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (5.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (6.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (7.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (8.2,1) circle (2cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you want to change the coloring of each circle, you can do this by adding the following color options.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [green] (1,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [pink] (1.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [cyan] (2.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [magenta] (3.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [red] (4.2,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue] (5,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [brown] (5.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [yellow] (6.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [purple] (7.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [orange] (8.2,1) circle (2cm);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe compilation results of adding the color options above are as follows.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEhXeMabl0FNkxe5EzxNCdTyqiEuAuqlfUKXHubCGfDnY20-Zv69Gt_twdvd0rs_BQ0gMh6dE5XHGq7uaJ9S9y71dt1iFvhMS4uw_bYC3faLkglC2lykEJ6yaQ9npEqDDfRzi8m-IEU4jlUR0k3fnb8ZIlnIG7ZilNwJu0h3OqV1MDHi_12hjHBHW657=s593\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"ilusi optik lingkaran di latex\" border=\"0\" data-original-height=\"283\" data-original-width=\"593\" height=\"153\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEhXeMabl0FNkxe5EzxNCdTyqiEuAuqlfUKXHubCGfDnY20-Zv69Gt_twdvd0rs_BQ0gMh6dE5XHGq7uaJ9S9y71dt1iFvhMS4uw_bYC3faLkglC2lykEJ6yaQ9npEqDDfRzi8m-IEU4jlUR0k3fnb8ZIlnIG7ZilNwJu0h3OqV1MDHi_12hjHBHW657=w320-h153\" title=\"ilusi optik lingkaran\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"complete-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete Code\"\u003e\u003c/a\u003eComplete Code\u003c/h2\u003e\u003cp\u003eIn addition, Below is the complete code for the circular optical illusion image.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[line width=1mm, blue]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (2.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (3.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (4.2,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (5,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (5.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (6.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (7.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (8.2,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd for colored circles with the following code.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 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class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[line width=1mm]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [green] (1,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [pink] (1.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [cyan] (2.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [magenta] (3.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [red] (4.2,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [blue] (5,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [brown] (5.8,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [yellow] (6.6,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [purple] (7.4,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\t\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e [orange] (8.2,1) circle (2cm);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\t\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eYou can compile and see the results using the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e engine available on this blog. Please paste the code here.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThese are the steps for drawing a circular optical illusion using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. In your opinion, does the circular optical illusion image resemble a tunnel or other shape? Of course, it comes back to your own vision. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Education Calendar for Academic Year 2021/2022", "date_published" : "2023-03-01T13:37:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/06/kalender-pendidikan-tahun-pelajaran/", "url" : "https://www.aantriono.com/2021/06/kalender-pendidikan-tahun-pelajaran/", "summary": "It doesn\u0026rsquo;t feel like we\u0026rsquo;ll be entering a new school year soon. Of course, the teacher does a lot of preparation. As we know, the teacher\u0026rsquo;s job is to plan, implement, evaluate, assess and follow up on learning outcomes. Regarding planning learning, teachers need to create a syllabus, lesson plans, teaching materials, LKPD and so on. For timing, refer to the educational calendar made by the district or provincial education and culture office. The educational calendar is a time setting for students\u0026rsquo; learning activities during one academic year which includes the beginning of the academic year, effective learning weeks, effective learning time and holidays.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eIt doesn\u0026rsquo;t feel like we\u0026rsquo;ll be entering a new school year soon. Of course, the teacher does a lot of preparation. As we know, the teacher\u0026rsquo;s job is to plan, implement, evaluate, assess and follow up on learning outcomes. Regarding planning learning, teachers need to create a syllabus, lesson plans, teaching materials, LKPD and so on. For timing, refer to the educational calendar made by the district or provincial education and culture office. The educational calendar is a time setting for students\u0026rsquo; learning activities during one academic year which includes the beginning of the academic year, effective learning weeks, effective learning time and holidays.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-pxhz786CIuPq9E4UErMyeWPKYIjwCQ7vEqsvxv6P7B7SWO-mYSuroBGITY-HFi_tki9q9FbHMrKbaKrke_AH8JNr4TqIR4KANC66og3rUGMONma0ZN0K6kBCMMeJeg5ooiMfXy6Ybbo/s651/Screenshot+2021-06-26+23.56.14.png' alt=\"kalender pendidikan TP 2021/2022 di LaTeX\" \u003e\u003c/p\u003e\n\u003ch3 id=\"educational-calendar-components\" class=\"headerLink\"\u003e\n \u003ca href=\"#educational-calendar-components\" class=\"header-mark\" aria-label=\"Direct link to section: Educational Calendar Components\"\u003e\u003c/a\u003eEducational Calendar Components\u003c/h3\u003e\u003cp\u003eThe start of the school year is the time when learning activities begin at the beginning of the school year in each educational unit. Effective learning weeks are the number of weeks of learning activities for each academic year in each educational unit. Effective learning time is the number of learning hours each week, including the number of learning hours for all subjects including local content, plus the number of hours for self-development activities. Holiday time is the time specified when scheduled learning activities are not held in the educational unit in question. Holiday time can take the form of mid-semester breaks, inter-semester breaks, end-of-school year holidays, religious holidays, public holidays including national holidays, and special holidays.\u003c/p\u003e\n\u003ch3 id=\"create-an-education-calendar\" class=\"headerLink\"\u003e\n \u003ca href=\"#create-an-education-calendar\" class=\"header-mark\" aria-label=\"Direct link to section: Create an Education Calendar\"\u003e\u003c/a\u003eCreate an Education Calendar\u003c/h3\u003e\u003cp\u003eIn this post, I will create a guide for creating an educational calendar using LaTeX. Of course, the syntax is different from educational calendars created using Microsoft Word or Microsoft Excel. I obtained the technicalities for creating an educational calendar using LaTeX \u003ca href=\"https://menulislatex.blogspot.com/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003e\u003cstrong\u003ehere\u003c/strong\u003e\u003c/a\u003e. Then I adjust it to the needs of the 2021/2022 academic year.\u003c/p\u003e\n\u003ch3 id=\"document-class\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-class\" class=\"header-mark\" aria-label=\"Direct link to section: Document Class\"\u003e\u003c/a\u003eDocument Class\u003c/h3\u003e\u003cp\u003eThe document class used to create an educational calendar is *scrartcl, *and then we write it using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[fontsize=20pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escrartcl\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003ehttp://tex.stackexchange.com/questions/170462/calendar-in-latex/170474#170474\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"preamble\" class=\"headerLink\"\u003e\n \u003ca href=\"#preamble\" class=\"header-mark\" aria-label=\"Direct link to section: Preamble\"\u003e\u003c/a\u003ePreamble\u003c/h3\u003e\u003cp\u003eIn the preamble, we will write the packages and orders needed to create the educational calendar. In the preamble section we can determine the size of the letters used, the color of the letters, and the size (width and length) of the boxes containing the date.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebookman\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[indonesian]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[utf8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003einputenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[T1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efontenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edejavu\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=1cm,a4paper,landscape]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[dvipsnames,svgnames]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pagestyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eempty\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalendar,fit\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eexpl3,xparse\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Adapted from http://tex.stackexchange.com/a/10199/4771\u003cspan class=\"k\"\u003e\\makeatletter\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzoption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday headings\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday heading\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[#1]\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday heading\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[]\u003cspan class=\"k\"\u003e\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday letter headings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[ execute before day scope=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifdate\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday of month=1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@ya\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@yshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\\pgf\u003c/span\u003e@xa\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@xshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformyshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@ya\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\d\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0/Senin,1/Selasa,2/Rabu,3/Kamis,\u003cspan class=\"err\"\u003e%\u003c/span\u003e 4/Jumat,5/Sabtu,6/Minggu\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa=\u003cspan class=\"k\"\u003e\\d\\pgf\u003c/span\u003e@xa\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformxshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e/2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformyshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@ya\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[above=-0.5ex,day heading]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}{}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e]\u003cspan class=\"k\"\u003e\\makeatother\u003c/span\u003e\u003cspan class=\"err\"\u003e%%%\u003c/span\u003e End\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Adapted from pgf source\u003cspan class=\"k\"\u003e\\def\\pgfcalendarmonthname\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\translate\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ifcase\u003c/span\u003e#1\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Januari\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Februari\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Maret\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e April\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Mei\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Juni\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Juli\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Agustus\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e September\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Oktober\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e November\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Desember\u003cspan class=\"k\"\u003e\\fi\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\makeatletter\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emonth label above centered\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[\u003cspan class=\"err\"\u003e%\u003c/span\u003e execute before day scope=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifdate\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday of month=1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@xshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb=\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@width\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\advance\\pgf\u003c/span\u003e@xb by-\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb=.5\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformxshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformxshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e/2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@y\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@yshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformyshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e0.333\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@y\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\tikzmonthcode\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}{}}\u003c/span\u003e, every month/.append style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eanchor=base\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\u003cspan class=\"k\"\u003e\\makeatother\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e End\u003cspan class=\"k\"\u003e\\ExplSyntaxOn\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Adapted from http://tex.stackexchange.com/a/56214/4771\u003cspan class=\"err\"\u003e%%%\u003c/span\u003e first of all we define the user level commands\u003cspan class=\"k\"\u003e\\NewDocumentCommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\addtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e m \u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\NewDocumentCommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\addtextyear\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e mm \u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd\u003cspan class=\"nb\"\u003e_\u003c/span\u003ey:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\NewDocumentCommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\showtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eoutput\u003cspan class=\"nb\"\u003e_\u003c/span\u003edirect: \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e allocate variable:\u003cspan class=\"err\"\u003e%\u003c/span\u003e a sequence for global storage of the inputs;\u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq\u003cspan class=\"err\"\u003e%\u003c/span\u003e store globally an input in the sequence\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:n #1 \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003egput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eright:Nn \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd\u003cspan class=\"nb\"\u003e_\u003c/span\u003ey:nn #1 #2 \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003egput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eright:Nn \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 ~ ( \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eto\u003cspan class=\"nb\"\u003e_\u003c/span\u003earabic:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e - #2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e ) \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e how to output in direct order; we simply do a mapping function calling\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprint:n after incrementing the counter\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprotected:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eoutput\u003cspan class=\"nb\"\u003e_\u003c/span\u003edirect: \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emap\u003cspan class=\"nb\"\u003e_\u003c/span\u003einline:Nn \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprint:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e ##1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003egclear:N \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e the printing macro; change here for adapting to your wishes\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprint:n #1 \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"k\"\u003e\\par\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e End\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Knuth\u0026#39;s AoCP, vol 1, 2nd ed, pp 155--156\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eZ\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eD\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Nn \u003cspan class=\"k\"\u003e\\easter\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 19 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 3 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - 12 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eZ\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 8 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 25 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eD\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 5 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 10 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 11 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 20 + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eZ\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 30 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn behaves strangely. \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 0 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 30 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 25 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026gt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 11 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eincr:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 24 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eincr:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 44 - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 21 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 30 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eD\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026gt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 31 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esub:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 31 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e April \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 3 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e March \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendardatetojulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Easter/.default = 0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Easter/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\easter\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdatejulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + #1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarmatchestrue\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Nn \u003cspan class=\"k\"\u003e\\advent\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendardatetojulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 12 - 25 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarjuliantoweekday\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 22 \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Advent/.default = 0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Advent/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\advent\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdatejulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + #1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarmatchestrue\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e http://www.tondering.dk/claus/cal/week.php#calcweekno\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ed\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Nn \u003cspan class=\"k\"\u003e\\week\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber:nnn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #3 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 3 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e jan or feb \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ezero:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 + 31 * ( \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 ) \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end true \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeq:NN \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 153 * ( \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 3 ) + 2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 58 + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ed\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 3 - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ed\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 0 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003etrue \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 53 - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end true \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026gt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 364 + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end true \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end false \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end false\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eroed\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.937254901961,0.16862745098,0.176470588235\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblaa\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0,0.156862745098,0.407843137255\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newsavebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\flagNO\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\savebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\flagNO\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed]\u003c/span\u003e rectangle (6pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed,yshift=10pt]\u003c/span\u003e rectangle (6pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed,xshift=10pt]\u003c/span\u003e rectangle (12pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed,xshift=10pt,yshift=10pt]\u003c/span\u003e rectangle (12pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blaa,yshift=7pt]\u003c/span\u003e rectangle (22pt,2pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blaa,xshift=7pt]\u003c/span\u003e rectangle (2pt,16pt);\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/week~number/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\week\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber:nnn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdatemonth\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateday\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\addtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003edapat diisi \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ExplSyntaxOff\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/flag-flying day/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e,0) node [above right,font=\u003cspan class=\"k\"\u003e\\Huge\u003c/span\u003e] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\resizebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e!\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.8ex\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\flagNO\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}}\u003c/span\u003e; \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/observance/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\addtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/anniversary/.code 2 args=\u003cspan class=\"k\"\u003e\\addtextyear\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#2\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/day code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e (lower right) at (0,0) [above left,font=\u003cspan class=\"k\"\u003e\\huge\u003c/span\u003e] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzdaytext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e (upper left) at (-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e) [below right,align=left,text width=\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\pgflinewidth\u003c/span\u003e, font=\u003cspan class=\"k\"\u003e\\tiny\u003c/span\u003e,blue] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\showtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e (lower left) at (-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e,0) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue,rounded corners, draw, fit=(lower right) (upper left) (lower left), inner sep=1mm]\u003c/span\u003e \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/inner sep = 0pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/day xshift=\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e+2mm+2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/day yshift=\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e+2mm+2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e20mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e25mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"document-body\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-body\" class=\"header-mark\" aria-label=\"Direct link to section: Document Body\"\u003e\u003c/a\u003eDocument Body\u003c/h3\u003e\u003cp\u003eNext we enter the body of the document and start writing the school identity, using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\centering\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eKalender Pendidikan\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eSMP/MTs ................\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eTahun Pelajaran 2021/2022\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\vspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e.5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"july-2021\" class=\"headerLink\"\u003e\n \u003ca href=\"#july-2021\" class=\"header-mark\" aria-label=\"Direct link to section: July 2021\"\u003e\u003c/a\u003eJuly 2021\u003c/h3\u003e\u003cp\u003eThe start of the 2021/2022 academic year starts in July 2021. The start of school starts on Monday, July 12 2021. For the following months, please adjust it to the needs of each school. For details, pay attention to the following commands.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-07-01 to 2021-07-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Monday, equals=01-07, equals=02-07, equals=03-07, equals=04-07, equals=05-07, equals=06-07, equals=07-07, equals=08-07, equals=09-07, equals=10-07, equals=11-07, equals=12-07) [week number] if (Sunday, equals=07-04) [red] if (Friday, equals=07-02) [SeaGreen] if (equals=07-05) [blue,observance=Milad] if (equals=07-12) [observance=HPS] if (equals=07-13) [observance=MPLS] if (equals=07-14) [observance=MPLS] if (equals=07-20) [observance=Hari Raya Idul Adha];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"august-2021\" class=\"headerLink\"\u003e\n \u003ca href=\"#august-2021\" class=\"header-mark\" aria-label=\"Direct link to section: August 2021\"\u003e\u003c/a\u003eAugust 2021\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-08-01 to 2021-08-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=08-01) [red] if (Friday, equals=08-06) [SeaGreen] if (equals=08-10) [red,observance= Tahun Baru Islam 1443 H] if (equals=08-17) [blue,observance=HUT RI];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"september-2021\" class=\"headerLink\"\u003e\n \u003ca href=\"#september-2021\" class=\"header-mark\" aria-label=\"Direct link to section: September 2021\"\u003e\u003c/a\u003eSeptember 2021\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-09-01 to 2021-09-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=09-05) [red] if (Friday, equals=09-03) [SeaGreen] if (equals=09-06) [blue,observance=PTS] if (equals=09-07) [blue,observance=PTS] if (equals=09-08) [blue,observance=PTS] if (equals=09-09) [blue,observance=PTS] if (equals=09-10) [blue,observance=PTS] if (equals=09-11) [blue,observance=PTS];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"october-2021\" class=\"headerLink\"\u003e\n \u003ca href=\"#october-2021\" class=\"header-mark\" aria-label=\"Direct link to section: October 2021\"\u003e\u003c/a\u003eOctober 2021\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-11-01 to 2021-11-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=11-07) [red] if (Friday, equals=11-05) [SeaGreen] if (equals=11-10) [blue,observance=Hari pahlawan] if (equals=11-25) [blue,observance=Hari guru];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"december-2021\" class=\"headerLink\"\u003e\n \u003ca href=\"#december-2021\" class=\"header-mark\" aria-label=\"Direct link to section: December 2021\"\u003e\u003c/a\u003eDecember 2021\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-12-01 to 2021-12-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=12-05) [red] if (Friday, equals=12-03) [SeaGreen] if (equals=12-06) [blue,observance=PAS] if (equals=12-07) [blue,observance=PAS] if (equals=12-08) [blue,observance=PAS] if (equals=12-09) [blue,observance=PAS] if (equals=12-10) [blue,observance=PAS] if (equals=12-11) [blue,observance=PAS] if (equals=12-18) [blue,observance=Penyerahan raport] if (equals=12-20) [red,observance=Libur semester ganjil] if (equals=12-21) [red,observance=Libur semester ganjil] if (equals=12-22) [red,observance=Libur semester ganjil] if (equals=12-23) [red,observance=Libur semester ganjil] if (equals=12-24) [red,observance=Libur semester ganjil] if (equals=12-25) [red,observance=Hari raya natal] if (equals=12-27) [red,observance=Libur semester ganjil] if (equals=12-28) [red,observance=Libur semester ganjil] if (equals=12-29) [red,observance=Libur semester ganjil] if (equals=12-30) [red,observance=Libur semester ganjil] if (equals=12-31) [red,observance=Libur semester ganjil];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"january-2022\" class=\"headerLink\"\u003e\n \u003ca href=\"#january-2022\" class=\"header-mark\" aria-label=\"Direct link to section: January 2022\"\u003e\u003c/a\u003eJanuary 2022\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-01-01 to 2022-01-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=01-02) [red] if (Friday, equals=01-07) [SeaGreen] if (equals=01-01) [red,observance=Tahun baru masehi 2022 M];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"february-2022\" class=\"headerLink\"\u003e\n \u003ca href=\"#february-2022\" class=\"header-mark\" aria-label=\"Direct link to section: February 2022\"\u003e\u003c/a\u003eFebruary 2022\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-02-01 to 2022-02-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=02-06) [red] if (Friday, equals=02-04) [SeaGreen] if (equals=02-01) [red,observance=Tahun baru imlek 2573];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"march-2022\" class=\"headerLink\"\u003e\n \u003ca href=\"#march-2022\" class=\"header-mark\" aria-label=\"Direct link to section: March 2022\"\u003e\u003c/a\u003eMarch 2022\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-03-01 to 2022-03-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=03-06) [red] if (Friday, equals=03-04) [SeaGreen] if (equals=03-01) [red,observance=Isra\u0026#39; mikraj] if (equals=03-03) [red,observance=Hari raya nyepi 1944 S] if (equals=03-14) [blue,observance=PTS] if (equals=03-15) [blue,observance=PTS] if (equals=03-16) [blue,observance=PTS] if (equals=03-17) [blue,observance=PTS] if (equals=03-18) [blue,observance=PTS] if (equals=03-19) [blue,observance=PTS];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"april-2022\" class=\"headerLink\"\u003e\n \u003ca href=\"#april-2022\" class=\"header-mark\" aria-label=\"Direct link to section: April 2022\"\u003e\u003c/a\u003eApril 2022\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-04-01 to 2022-04-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=04-03) [red] if (Friday, equals=04-01) [SeaGreen] if (equals=04-15) [red,observance=Wafat Isa al masih] if (equals=04-17) [red,observance=Hari paskah] if (equals=04-18) [blue,observance=Ujian Sekolah] if (equals=04-19) [blue,observance=Ujian Sekolah] if (equals=04-20) [blue,observance=Ujian Sekolah] if (equals=04-21) [blue,observance=Ujian Sekolah] if (equals=04-22) [blue,observance=Ujian Sekolah] if (equals=04-23) [blue,observance=Ujian Sekolah] if (equals=04-28) [red,observance=Libur Hari Raya] if (equals=04-29) [red,observance=Libur Hari Raya] if (equals=04-30) [red,observance=Libur Hari Raya];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"may-2022\" class=\"headerLink\"\u003e\n \u003ca href=\"#may-2022\" class=\"header-mark\" aria-label=\"Direct link to section: May 2022\"\u003e\u003c/a\u003eMay 2022\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-14\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-05-01 to 2022-05-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=05-01) [red] if (Friday, equals=05-06) [SeaGreen] if (equals=05-01) [red,observance=Hari buruh] if (equals=05-02) [red,observance=Libur Hari Raya] if (equals=05-03) [red,observance=Hari raya idul fitri 1443 H] if (equals=05-04) [red,observance=Hari raya idul fitri 1443 H] if (equals=05-05) [red,observance=Libur Hari Raya] if (equals=05-06) [red,observance=Libur Hari Raya] if (equals=05-07) [red,observance=Libur Hari Raya] if (equals=05-08) [red,observance=Libur Hari Raya] if (equals=05-09) [red,observance=Libur Hari Raya] if (equals=05-16) [red,observance=Hari raya waisak]if (equals=05-26) [red,observance=Kenaikan Isa al masih] ;\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch3 id=\"june-2022\" class=\"headerLink\"\u003e\n \u003ca href=\"#june-2022\" class=\"header-mark\" aria-label=\"Direct link to section: June 2022\"\u003e\u003c/a\u003eJune 2022\u003c/h3\u003e\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-15\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-06-01 to 2022-06-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=06-05) [red] if (Friday, equals=06-03) [SeaGreen] if (equals=06-01) [red,observance=Hari lahir Pancasila] if (equals=06-06) [blue,observance=PAT] if (equals=06-07) [blue,observance=PAT] if (equals=06-08) [blue,observance=PAT] if (equals=06-09) [blue,observance=PAT] if (equals=06-10) [SeaGreen,observance=PAT] if (equals=06-11) [blue,observance=PAT] if (equals=06-22) [red,observance=Libur semester genap];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eLastly we will create text for the place, name and signature of the principal of the school where we teach, here I place it on the right, as well as to close the document using the following command.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-16\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushright\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e.........,\u003cspan class=\"k\"\u003e\\hspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e.8cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003eJuli 2021\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eKepala SMP/MTs .............\u003cspan class=\"k\"\u003e\\\\\\vspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1.5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e(.........................)\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eNIP ......................\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushright\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd now to see the compiled results of the documents we have written, please \u003ca href=\"https://drive.google.com/file/d/1tsMp0weTL8LJ2KXvtCPUyu5vuRxX5v57m/view?usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\"\u003edownload here\u003c/a\u003e.\u003c/p\u003e\n\u003ch3 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h3\u003e\u003cp\u003eStep by step has been taken to create an educational calendar using LaTeX, and you have seen the results. And to better understand, Below is the complete LaTeX code to create the educational calendar.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-17\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[fontsize=20pt]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escrartcl\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003ehttp://tex.stackexchange.com/questions/170462/calendar-in-latex/170474#170474\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebookman\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[indonesian]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebabel\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[utf8]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003einputenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[T1]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003efontenc\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edejavu\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[margin=1cm,a4paper,landscape]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egeometry\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[dvipsnames,svgnames]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pagestyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eempty\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003egraphicx\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ecalendar,fit\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eexpl3,xparse\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Adapted from http://tex.stackexchange.com/a/10199/4771\u003cspan class=\"k\"\u003e\\makeatletter\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzoption\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday headings\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday heading\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[#1]\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday heading\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[]\u003cspan class=\"k\"\u003e\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday letter headings\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[ execute before day scope=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifdate\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday of month=1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@ya\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@yshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\\pgf\u003c/span\u003e@xa\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@xshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformyshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@ya\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\d\u003c/span\u003e/\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e0/Senin,1/Selasa,2/Rabu,3/Kamis,\u003cspan class=\"err\"\u003e%\u003c/span\u003e 4/Jumat,5/Sabtu,6/Minggu\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa=\u003cspan class=\"k\"\u003e\\d\\pgf\u003c/span\u003e@xa\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformxshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e/2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformyshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@ya\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[above=-0.5ex,day heading]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e;\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}{}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e]\u003cspan class=\"k\"\u003e\\makeatother\u003c/span\u003e\u003cspan class=\"err\"\u003e%%%\u003c/span\u003e End\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Adapted from pgf source\u003cspan class=\"k\"\u003e\\def\\pgfcalendarmonthname\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\translate\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ifcase\u003c/span\u003e#1\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Januari\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Februari\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Maret\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e April\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Mei\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Juni\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Juli\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Agustus\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e September\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Oktober\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e November\u003cspan class=\"k\"\u003e\\or\u003c/span\u003e Desember\u003cspan class=\"k\"\u003e\\fi\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\makeatletter\\tikzstyle\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003emonth label above centered\u003cspan class=\"nb\"\u003e}\u003c/span\u003e=[\u003cspan class=\"err\"\u003e%\u003c/span\u003e execute before day scope=\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\ifdate\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eday of month=1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@xshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb=\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@width\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\advance\\pgf\u003c/span\u003e@xb by-\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xa\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb=.5\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformxshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@xb\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformxshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e/2\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfmathsetlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@y\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikz\u003c/span\u003e@lib@cal@yshift\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgftransformyshift\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e0.333\u003cspan class=\"k\"\u003e\\pgf\u003c/span\u003e@y\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\tikzmonthcode\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}{}}\u003c/span\u003e, every month/.append style=\u003cspan class=\"nb\"\u003e{\u003c/span\u003eanchor=base\u003cspan class=\"nb\"\u003e}\u003c/span\u003e]\u003cspan class=\"k\"\u003e\\makeatother\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e End\u003cspan class=\"k\"\u003e\\ExplSyntaxOn\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Adapted from http://tex.stackexchange.com/a/56214/4771\u003cspan class=\"err\"\u003e%%%\u003c/span\u003e first of all we define the user level commands\u003cspan class=\"k\"\u003e\\NewDocumentCommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\addtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e m \u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\NewDocumentCommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\addtextyear\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e mm \u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd\u003cspan class=\"nb\"\u003e_\u003c/span\u003ey:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\NewDocumentCommand\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\showtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"nb\"\u003e}{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eoutput\u003cspan class=\"nb\"\u003e_\u003c/span\u003edirect: \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e allocate variable:\u003cspan class=\"err\"\u003e%\u003c/span\u003e a sequence for global storage of the inputs;\u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq\u003cspan class=\"err\"\u003e%\u003c/span\u003e store globally an input in the sequence\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:n #1 \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003egput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eright:Nn \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd\u003cspan class=\"nb\"\u003e_\u003c/span\u003ey:nn #1 #2 \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003egput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eright:Nn \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 ~ ( \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eto\u003cspan class=\"nb\"\u003e_\u003c/span\u003earabic:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e - #2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e ) \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e how to output in direct order; we simply do a mapping function calling\u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprint:n after incrementing the counter\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprotected:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eoutput\u003cspan class=\"nb\"\u003e_\u003c/span\u003edirect: \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emap\u003cspan class=\"nb\"\u003e_\u003c/span\u003einline:Nn \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprint:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e ##1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\seq\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003egclear:N \u003cspan class=\"k\"\u003e\\g\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ebdaycal\u003cspan class=\"nb\"\u003e_\u003c/span\u003einput\u003cspan class=\"nb\"\u003e_\u003c/span\u003eseq \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003e the printing macro; change here for adapting to your wishes\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Npn \u003cspan class=\"k\"\u003e\\bdaycal\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eprint:n #1 \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"k\"\u003e\\par\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e End\u003cspan class=\"err\"\u003e%%\u003c/span\u003e Knuth\u0026#39;s AoCP, vol 1, 2nd ed, pp 155--156\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eZ\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eD\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Nn \u003cspan class=\"k\"\u003e\\easter\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 19 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 3 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - 12 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eZ\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 8 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eC\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 25 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eD\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 5 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 10 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 11 * \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 20 + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eZ\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eX\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 30 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn behaves strangely. \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 0 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 30 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 25 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eG\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026gt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 11 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eincr:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 24 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eincr:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 44 - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eE\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 21 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 30 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadd:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eD\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026gt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 31 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esub:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 31 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e April \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 3 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e March \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendardatetojulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eM\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003eN\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Easter/.default = 0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Easter/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\easter\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdatejulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeaster\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + #1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarmatchestrue\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Nn \u003cspan class=\"k\"\u003e\\advent\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendardatetojulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003eY\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 12 - 25 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarjuliantoweekday\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003exmas\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 22 \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Advent/.default = 0\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/pgf/calendar/Advent/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\advent\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003esunday:n \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnT \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdatejulian\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eadvent\u003cspan class=\"nb\"\u003e_\u003c/span\u003ejulian\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + #1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarmatchestrue\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"err\"\u003e%%\u003c/span\u003e http://www.tondering.dk/claus/cal/week.php#calcweekno\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ed\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint\u003cspan class=\"k\"\u003e\\cs\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enew:Nn \u003cspan class=\"k\"\u003e\\week\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber:nnn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e #3 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 3 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e jan or feb \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ezero:N \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 + 31 * ( \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 ) \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end true \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset\u003cspan class=\"nb\"\u003e_\u003c/span\u003eeq:NN \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eyear\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 4 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 100 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 400 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ec\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eday\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 153 * ( \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003emonth\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - 3 ) + 2 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 58 + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ea\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eb\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ed\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003emod:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ee\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ef\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint + 3 - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003ed\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026lt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 0 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003etrue \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 53 - \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eg\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint - \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 5 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end true \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ecompare:nNnTF \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u0026gt; \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 364 + \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003es\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e true \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end true \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e false \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eset:Nn \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003eW\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\int\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003ediv\u003cspan class=\"nb\"\u003e_\u003c/span\u003etruncate:nn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\l\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003eweek\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber\u003cspan class=\"nb\"\u003e_\u003c/span\u003en\u003cspan class=\"nb\"\u003e_\u003c/span\u003eint \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e 7 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e + 1 \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end false \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"err\"\u003e%\u003c/span\u003e end false\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eroed\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.937254901961,0.16862745098,0.176470588235\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblaa\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0,0.156862745098,0.407843137255\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newsavebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\flagNO\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\savebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\flagNO\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed]\u003c/span\u003e rectangle (6pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed,yshift=10pt]\u003c/span\u003e rectangle (6pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed,xshift=10pt]\u003c/span\u003e rectangle (12pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[roed,xshift=10pt,yshift=10pt]\u003c/span\u003e rectangle (12pt,6pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blaa,yshift=7pt]\u003c/span\u003e rectangle (22pt,2pt); \u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[blaa,xshift=7pt]\u003c/span\u003e rectangle (2pt,16pt);\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/week~number/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\week\u003c/span\u003e\u003cspan class=\"nb\"\u003e_\u003c/span\u003enumber:nnn \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateyear\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdatemonth\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\pgfcalendarifdateday\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"k\"\u003e\\addtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"err\"\u003e%\u003c/span\u003edapat diisi \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\ExplSyntaxOff\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/flag-flying day/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e,0) node [above right,font=\u003cspan class=\"k\"\u003e\\Huge\u003c/span\u003e] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\resizebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e!\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0.8ex\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usebox\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\flagNO\u003c/span\u003e\u003cspan class=\"nb\"\u003e}}}\u003c/span\u003e; \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/observance/.code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\addtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/anniversary/.code 2 args=\u003cspan class=\"k\"\u003e\\addtextyear\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e#1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e#2\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/day code = \u003cspan class=\"nb\"\u003e{\u003c/span\u003e \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e (lower right) at (0,0) [above left,font=\u003cspan class=\"k\"\u003e\\huge\u003c/span\u003e] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\tikzdaytext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e (upper left) at (-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e,\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e) [below right,align=left,text width=\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e-\u003cspan class=\"k\"\u003e\\pgflinewidth\u003c/span\u003e, font=\u003cspan class=\"k\"\u003e\\tiny\u003c/span\u003e,blue] \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\showtext\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e; \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e (lower left) at (-\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e,0) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \u003cspan class=\"k\"\u003e\\node\u003c/span\u003e\u003cspan class=\"na\"\u003e[blue,rounded corners, draw, fit=(lower right) (upper left) (lower left), inner sep=1mm]\u003c/span\u003e \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e; \u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/inner sep = 0pt\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/day xshift=\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e+2mm+2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\pgfkeys\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e/tikz/day yshift=\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e+2mm+2mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellheight\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e20mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\setlength\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\cellwidth\u003c/span\u003e\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e25mm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\centering\\textcolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eblue\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e\u003cspan class=\"k\"\u003e\\textbf\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eKalender Pendidikan\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eSMP/MTs ..............\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eTahun Pelajaran 2021/2022\u003cspan class=\"nb\"\u003e}}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\vspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e.5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-07-01 to 2021-07-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Monday, equals=01-07, equals=02-07, equals=03-07, equals=04-07, equals=05-07, equals=06-07, equals=07-07, equals=08-07, equals=09-07, equals=10-07, equals=11-07, equals=12-07) [week number] if (Sunday, equals=07-04) [red] if (Friday, equals=07-02) [SeaGreen] if (equals=07-05) [blue,observance=Milad] if (equals=07-12) [observance=HPS] if (equals=07-13) [observance=MPLS] if (equals=07-14) [observance=MPLS] if (equals=07-20) [red,observance=Hari Raya Idul Adha];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-08-01 to 2021-08-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=08-01) [red] if (Friday, equals=08-06) [SeaGreen] if (equals=08-10) [red,observance= Tahun Baru Islam 1443 H] if (equals=08-17) [blue,observance=HUT RI];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-09-01 to 2021-09-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=09-05) [red] if (Friday, equals=09-03) [SeaGreen] if (equals=09-06) [blue,observance=PTS] if (equals=09-07) [blue,observance=PTS] if (equals=09-08) [blue,observance=PTS] if (equals=09-09) [blue,observance=PTS] if (equals=09-10) [blue,observance=PTS] if (equals=09-11) [blue,observance=PTS];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-10-01 to 2021-10-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=10-03) [red] if (Friday, equals=10-08) [SeaGreen] if (equals=10-01) [blue,observance=Hari kesaktian Pancasila] if (equals=10-02) [blue,observance=Hari batik] if (equals=10-19) [red,observance=Maulid Nabi Muhammad SAW] if (equals=10-28) [blue,observance=Hari sumpah pemuda];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-11-01 to 2021-11-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=11-07) [red] if (Friday, equals=11-05) [SeaGreen] if (equals=11-10) [blue,observance=Hari pahlawan] if (equals=11-25) [blue,observance=Hari guru];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2021-12-01 to 2021-12-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=12-05) [red] if (Friday, equals=12-03) [SeaGreen] if (equals=12-06) [blue,observance=PAS] if (equals=12-07) [blue,observance=PAS] if (equals=12-08) [blue,observance=PAS] if (equals=12-09) [blue,observance=PAS] if (equals=12-10) [blue,observance=PAS] if (equals=12-11) [blue,observance=PAS] if (equals=12-18) [blue,observance=Penyerahan raport] if (equals=12-20) [red,observance=Libur semester ganjil] if (equals=12-21) [red,observance=Libur semester ganjil] if (equals=12-22) [red,observance=Libur semester ganjil] if (equals=12-23) [red,observance=Libur semester ganjil] if (equals=12-24) [red,observance=Libur semester ganjil] if (equals=12-25) [red,observance=Hari raya natal] if (equals=12-27) [red,observance=Libur semester ganjil] if (equals=12-28) [red,observance=Libur semester ganjil] if (equals=12-29) [red,observance=Libur semester ganjil] if (equals=12-30) [red,observance=Libur semester ganjil] if (equals=12-31) [red,observance=Libur semester ganjil];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-01-01 to 2022-01-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=01-02) [red] if (Friday, equals=01-07) [SeaGreen] if (equals=01-01) [red,observance=Tahun baru masehi 2022 M];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-02-01 to 2022-02-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=02-06) [red] if (Friday, equals=02-04) [SeaGreen] if (equals=02-01) [red,observance=Tahun baru imlek 2573];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-04-01 to 2022-04-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=04-03) [red] if (Friday, equals=04-01) [SeaGreen] if (equals=04-15) [red,observance=Wafat Isa al masih] if (equals=04-17) [red,observance=Hari paskah] if (equals=04-18) [blue,observance=Ujian Sekolah] if (equals=04-19) [blue,observance=Ujian Sekolah] if (equals=04-20) [blue,observance=Ujian Sekolah] if (equals=04-21) [blue,observance=Ujian Sekolah] if (equals=04-22) [blue,observance=Ujian Sekolah] if (equals=04-23) [blue,observance=Ujian Sekolah] if (equals=04-28) [red,observance=Libur Hari Raya] if (equals=04-29) [red,observance=Libur Hari Raya] if (equals=04-30) [red,observance=Libur Hari Raya];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-05-01 to 2022-05-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=05-01) [red] if (Friday, equals=05-06) [SeaGreen] if (equals=05-01) [red,observance=Hari buruh] if (equals=05-02) [red,observance=Libur Hari Raya] if (equals=05-03) [red,observance=Hari raya idul fitri 1443 H] if (equals=05-04) [red,observance=Hari raya idul fitri 1443 H] if (equals=05-05) [red,observance=Libur Hari Raya] if (equals=05-06) [red,observance=Libur Hari Raya] if (equals=05-07) [red,observance=Libur Hari Raya] if (equals=05-08) [red,observance=Libur Hari Raya] if (equals=05-09) [red,observance=Libur Hari Raya] if (equals=05-16) [red,observance=Hari raya waisak]if (equals=05-26) [red,observance=Kenaikan Isa al masih] ;\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\newpage\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[thick] \u003cspan class=\"k\"\u003e\\calendar\u003c/span\u003e\u003cspan class=\"na\"\u003e[dates=2022-06-01 to 2022-06-last, week list, month label above centered, month text=\\textsc{\\%mt \\%y0}, day headings={font=\\footnotesize}, day letter headings]\u003c/span\u003e if (Sunday, equals=06-05) [red] if (Friday, equals=06-03) [SeaGreen] if (equals=06-01) [red,observance=Hari lahir Pancasila] if (equals=06-06) [blue,observance=PAT] if (equals=06-07) [blue,observance=PAT] if (equals=06-08) [blue,observance=PAT] if (equals=06-09) [blue,observance=PAT] if (equals=06-10) [SeaGreen,observance=PAT] if (equals=06-11) [blue,observance=PAT] if (equals=06-22) [red,observance=Libur semester genap];\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushright\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e.........,\u003cspan class=\"k\"\u003e\\hspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e.8cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003eJuli 2021\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eKepala SMP/MTs .............\u003cspan class=\"k\"\u003e\\\\\\vspace\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1.5cm\u003cspan class=\"nb\"\u003e}\u003c/span\u003e(.........................)\u003cspan class=\"k\"\u003e\\\\\u003c/span\u003eNIP ......................\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eflushright\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eIf you want to change some parts of this educational calendar, I recommend doing it via a TeX \u0026rsquo;engine\u0026rsquo; such as TeXlive or online using the overleaf site. How to? Please read the guide \u003ca href=\"https://www.silagebra.com/2021/03/cara-menulis-latex-menggunakan-overleaf.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003e\u003cstrong\u003ehere\u003c/strong\u003e\u003c/a\u003e. That\u0026rsquo;s the article about creating an educational calendar for the 2021/2022 school year using LaTeX. If there are errors, please correct them, if you have input, please write them here. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Drawing Turkish Flag", "date_published" : "2023-03-01T13:35:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-bendera-turki/", "url" : "https://www.aantriono.com/en/menggambar-bendera-turki/", "summary": " Drawing Turkish Flag in LaTeXScure everything, this article is still related to the previous article about draw the moon and stars.\nAfter drawing the moon and stars, I remembered the Turkish flag, which apparently contains a crescent moon and stars. Finally I tried to draw the Turkish flag using LaTeX. Thank God it was successful and finally I had time to make a guide for Sira friends. Look at the following image.\nThere is a box that says Turkish Flag, then a red rectangle containing a crescent moon and a star, and the position of the star is tilted to the left. Oh yes, from now on I will use the pronoun friend sira for all loyal readers of silagebra.com. Sobat sira is an acronym for silagebra friends 😉. Hope you like it, heeee\u0026hellip;.OK, let\u0026rsquo;s just discuss how to draw the Turkish flag using LaTeX. Prepare a cup of fragrant coffee ☕ and also rengginang to accompany our blogging, heee..\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-turkish-flag-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-turkish-flag-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Turkish Flag in LaTeX\"\u003e\u003c/a\u003eDrawing Turkish Flag in LaTeX\u003c/h2\u003e\u003cp\u003eScure everything, this article is still related to the previous article about \u003ca href=\"https://www.aantriono.com/2021/06/bulan-dan-bintang.html\" target=\"_blank\"\u003edraw the moon and stars.\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eAfter drawing the moon and stars, I remembered the Turkish flag, which apparently contains a crescent moon and stars. Finally I tried to draw the Turkish flag using LaTeX. Thank God it was successful and finally I had time to make a guide for Sira friends. Look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjX3TK5zSHd82oviP6lRNqOWnSc1THoIHNtEgWMPGy3Sc2dk4GsOFxTK2yG7OhA_WJfLYYoGnRmz4Ugun4qNHV_Tsq0Kz2m9cMVT051FilyoQzdRZR0pP2tfkT-HqGD1Wrn-yqLWJeqA7M/s637/Screenshot+2021-06-23+10.51.09.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bendera turki di latex\" border=\"0\" data-original-height=\"412\" data-original-width=\"637\" height=\"207\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjX3TK5zSHd82oviP6lRNqOWnSc1THoIHNtEgWMPGy3Sc2dk4GsOFxTK2yG7OhA_WJfLYYoGnRmz4Ugun4qNHV_Tsq0Kz2m9cMVT051FilyoQzdRZR0pP2tfkT-HqGD1Wrn-yqLWJeqA7M/w320-h207/Screenshot+2021-06-23+10.51.09.png\" title=\"bendera Turki\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThere is a box that says Turkish Flag, then a red rectangle containing a crescent moon and a star, and the position of the star is tilted to the left. Oh yes, from now on I will use the pronoun friend sira for all loyal readers of silagebra.com. Sobat sira is an acronym for silagebra friends 😉. Hope you like it, heeee\u0026hellip;.OK, let\u0026rsquo;s just discuss how to draw the Turkish flag using LaTeX. Prepare a cup of fragrant coffee ☕ and also rengginang to accompany our blogging, heee..\u003c/p\u003e\n\u003c!--more--\u003e\n\u003ch2 id=\"step-by-step-tikz-construction\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-tikz-construction\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step TikZ Construction\"\u003e\u003c/a\u003eStep-by-Step TikZ Construction\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen your text editor, friend.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSelect the article document class and type the following command.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\documentclass{article}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eDetermine the packages needed for drawing by typing the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\usepackage{fancybox}\\usepackage[x11names,dvipsnames]{xcolor}\\definecolor{bulanbintang}{rgb} {255,255,255}\\usepackage{tikz}\\usetikzlibrary{shapes.geometric}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eIf you want to provide the title, author, and creation date of the document, use the following command in the preamble.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\title{Bendera Turki}\\author{Aan Triono }\\date{\\today}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eEntering the body of the document, type the following command to display the title, author and date of creation of the document at the same time so that the flag is positioned in the middle of the page.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\begin{document}\\centering\\maketitle\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eNext, to create a box containing the Turkish Flag text, you can create it by writing the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\huge \\textcolor{red}{\\shadowbox{Bendera Turki}}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eThen from the image above you can see that there is empty space/space between the text box and the flag image, so we write the vertical space command as follows, while also giving the command to start drawing.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\vspace{0.5cm}\\begin{tikzpicture}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eThen we draw a red rectangular box using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill [red] (-5,-4) rectangle (8.5,3.5);\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eUp to this step, after you have compiled it, the results look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKxEtDfgrLJxf0hhS5a3TUDOBWgGSVoA_hGQaZM8OoyMQGaCT5ofXs1yW87Uo8ksooj_Ug2mn1VYj9iCtd-MZ3ngj7G1m9zUablIuyGGPkR27FDi6SGCnEcb6uKi86MEkjgKKzncQRTG8/s1268/Screenshot+2021-06-23+11.11.17.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bendera turki menggunakan latex\" border=\"0\" data-original-height=\"447\" data-original-width=\"1268\" height=\"113\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKxEtDfgrLJxf0hhS5a3TUDOBWgGSVoA_hGQaZM8OoyMQGaCT5ofXs1yW87Uo8ksooj_Ug2mn1VYj9iCtd-MZ3ngj7G1m9zUablIuyGGPkR27FDi6SGCnEcb6uKi86MEkjgKKzncQRTG8/w320-h113/Screenshot+2021-06-23+11.11.17.png\" title=\"bendera turki\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"9\"\u003e\n\u003cli\u003eThen we start drawing a white circle as a first step in drawing a crescent moon using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[bulanbintang,even odd rule] (0,0) circle (2.5);\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThe compilation results will look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6nQPffkHAqT8PC0Wt9kKjKQL2eX9T70KdpR0q13_TM1X7EdrgSfx5KGODpWKRmE9BTtHx9BoxAoYbPK5egx89xyMVZO6Sg3XVNPYMKW44CjsMFuggClFYdF3VcSgGUz8JlAw5s4W_azw/s1253/Screenshot+2021-06-23+11.20.48.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"lingkaran putih di latex\" border=\"0\" data-original-height=\"396\" data-original-width=\"1253\" height=\"101\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6nQPffkHAqT8PC0Wt9kKjKQL2eX9T70KdpR0q13_TM1X7EdrgSfx5KGODpWKRmE9BTtHx9BoxAoYbPK5egx89xyMVZO6Sg3XVNPYMKW44CjsMFuggClFYdF3VcSgGUz8JlAw5s4W_azw/w320-h101/Screenshot+2021-06-23+11.20.48.png\" title=\"lingkaran putih\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"10\"\u003e\n\u003cli\u003eThen we draw another second red circle, so that the first circle will change shape to a crescent moon using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[red] (.54,0) circle (2);\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThe compilation results will look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEii3ho8Zvm7-YZcNQjKNR6vWOJ0VJ-12t__UbrJAKrYsRc5EueSVqbTnCHfqkl92QodN0uH9ni5EFWWh5I-XxcB5hd0fFJ9PvajvRP25ke7LQkdEFaboJyfclknRiKu933G6TeCjKLHXbI/s1253/Screenshot+2021-06-23+11.25.10.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bulan sabit menggunakan latex\" border=\"0\" data-original-height=\"431\" data-original-width=\"1253\" height=\"110\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEii3ho8Zvm7-YZcNQjKNR6vWOJ0VJ-12t__UbrJAKrYsRc5EueSVqbTnCHfqkl92QodN0uH9ni5EFWWh5I-XxcB5hd0fFJ9PvajvRP25ke7LQkdEFaboJyfclknRiKu933G6TeCjKLHXbI/w320-h110/Screenshot+2021-06-23+11.25.10.png\" title=\"bulan sabit\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"11\"\u003e\n\u003cli\u003eThen to draw stars, we use the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[bulanbintang] (2.9,0) node [ regular polygon, regular polygon sides=5, inner sep=0pt, minimum size=1.5cm, ] (A) {} (A.corner 1)\\foreach \\i in{1, 3, 5, 2, 4} {-- (A.corner \\i) } -- cycle;\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThe compilation results will look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcoq1qHFI_Di0Vb07uttSevz956Bx66Heb_TrFic7yjZ_-O_Wxjt5DT0VkgRpvO58IFqGrwvY2Rfehv1vggWnphxnoVkfuUJPM7scdc144CNn_YPCova6VPOcifzA84pA2OZG5rOmTBOo/s1253/Screenshot+2021-06-23+11.27.13.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bendera turki menggunakan latex\" border=\"0\" data-original-height=\"431\" data-original-width=\"1253\" height=\"110\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcoq1qHFI_Di0Vb07uttSevz956Bx66Heb_TrFic7yjZ_-O_Wxjt5DT0VkgRpvO58IFqGrwvY2Rfehv1vggWnphxnoVkfuUJPM7scdc144CNn_YPCova6VPOcifzA84pA2OZG5rOmTBOo/w320-h110/Screenshot+2021-06-23+11.27.13.png\" title=\"bendera turki\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"12\"\u003e\n\u003cli\u003eFrom the image in step 11, you can see that the position of the star is not tilted to the left. To tilt it we use the following command and put it in the \\node option.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003eshape border rotate=20\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"13\"\u003e\n\u003cli\u003eTo conclude the drawing and document commands, we use the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eHow are you, sir? Are the compilation results the same as the first image in this article? If not, please repeat the steps and pay attention to the instructions. Don\u0026rsquo;t miss anything or make a mistake in writing it. Thus the article on drawing the Turkish flag using LaTeX. If there are errors, please correct them, if you have input, please write it here and if you like it, please follow this blog ❤. I hope this is helpful.\u003c/p\u003e", "language": "en" }, { "title" : "Moon and Stars", "date_published" : "2023-03-01T13:30:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bulan-dan-bintang/", "url" : "https://www.aantriono.com/en/bulan-dan-bintang/", "summary": " Drawing Moon and Stars in LaTeXOne of the strengths of LaTeX that cannot be doubted is its ability to draw certain objects with great precision. Of course, this is supported by the use of the TikZ package and its libraries. By determining the coordinates of the image precisely and precisely, you will get a stunning image (at least in my view). The following image is no exception.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-moon-and-stars-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-moon-and-stars-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Moon and Stars in LaTeX\"\u003e\u003c/a\u003eDrawing Moon and Stars in LaTeX\u003c/h2\u003e\u003cp\u003eOne of the strengths of LaTeX that cannot be doubted is its ability to draw certain objects with great precision. Of course, this is supported by the use of the TikZ package and its libraries. By determining the coordinates of the image precisely and precisely, you will get a stunning image (at least in my view). The following image is no exception.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiVvQkGzZiQfDW6PQlv8VRomzq6uR1-sHEZojNrGr7DJmdM8s5hSUqxU9sZqWRPaUk16AjSsTZsBbSqGeAJXqnBIAPM5nZbgzWBP3XeE6KJIQ-t4H8nnKtMBGLrSpOcefYFdVmQfdBLbI/s445/Screenshot+2021-06-11+21.19.13.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bulan dan bintang di latex\" border=\"0\" data-original-height=\"445\" data-original-width=\"310\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiVvQkGzZiQfDW6PQlv8VRomzq6uR1-sHEZojNrGr7DJmdM8s5hSUqxU9sZqWRPaUk16AjSsTZsBbSqGeAJXqnBIAPM5nZbgzWBP3XeE6KJIQ-t4H8nnKtMBGLrSpOcefYFdVmQfdBLbI/w223-h320/Screenshot+2021-06-11+21.19.13.png\" title=\"bulan dan bintang\" width=\"223\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eOf course, the charm here is relative in nature, in accordance with each personal\u0026rsquo;s taste values, which are of course different. But at least the images produced by LaTeX are very precise. OK, let\u0026rsquo;s just discuss how to draw the moon and stars using LaTeX.\u003c/p\u003e\n\u003ch2 id=\"step-by-step-tikz-construction\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-tikz-construction\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step TikZ Construction\"\u003e\u003c/a\u003eStep-by-Step TikZ Construction\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen your LaTeX text editor, in this case I use the overleaf.com site.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThe document class used is article by using the following command.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003earticle\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eNext, the packages required are as follows.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"na\"\u003e[x11names,dvipsnames]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003excolor\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\usetikzlibrary\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003eshapes.geometric\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eFor coloring, you can use the following color code.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\definecolor\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebulanbintang\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ergb\u003cspan class=\"nb\"\u003e}{\u003c/span\u003e0,0,205\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eThen we start drawing a blue circle according to the selected coloring by writing the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[bulanbintang,even odd rule]\u003c/span\u003e (0,0) circle (2.5);\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result looks like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWsMx5uIUN3WiUp_5cEgWELllwFNQlZJZEARtxKo8Cf9Brm9SQdn3mka1yDso8m2q_5X70CULNRA0OmpWLRc_7-PTjYShBWv0ykmkLDItcGZZS5ZzmBfY_8mM2eT-rQbsRlAJNi-Lgm08/s936/Screenshot+2021-06-11+21.43.12.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bulan dan bintang di latex\" border=\"0\" data-original-height=\"249\" data-original-width=\"936\" height=\"85\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWsMx5uIUN3WiUp_5cEgWELllwFNQlZJZEARtxKo8Cf9Brm9SQdn3mka1yDso8m2q_5X70CULNRA0OmpWLRc_7-PTjYShBWv0ykmkLDItcGZZS5ZzmBfY_8mM2eT-rQbsRlAJNi-Lgm08/w320-h85/Screenshot+2021-06-11+21.43.12.png\" title=\"bulan dan bintang\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eNext we will create a white circle inside the first circle so that the first circle is shaped like a crescent moon, the instructions are as follows.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[white]\u003c/span\u003e (0,.54) circle (2);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result looks like the following image\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7nUdDXq7GrNOGd2t1ZwmzDi0vSLeoI0k9qFgA_bCZvAfSmfng-K4NG2FZPJ-xDuqDKmIOtpDdOlRkMftYnN_yMSUwGWWV8_kKYeWeQwEW0DaGLJIg_WvFL8tmaOz2gI9ufwBa1Nq6ah4/s936/Screenshot+2021-06-11+21.46.38.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bulan dan bintang di latex\" border=\"0\" data-original-height=\"279\" data-original-width=\"936\" height=\"95\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7nUdDXq7GrNOGd2t1ZwmzDi0vSLeoI0k9qFgA_bCZvAfSmfng-K4NG2FZPJ-xDuqDKmIOtpDdOlRkMftYnN_yMSUwGWWV8_kKYeWeQwEW0DaGLJIg_WvFL8tmaOz2gI9ufwBa1Nq6ah4/w320-h95/Screenshot+2021-06-11+21.46.38.png\" title=\"bulan dan bintang\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eNext, we will draw a star right in the middle of the curve of the crescent moon, using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\fill\u003c/span\u003e\u003cspan class=\"na\"\u003e[bulanbintang]\u003c/span\u003e (0,2.5) node [regular polygon, regular polygon sides=5, inner sep=0pt, minimum size=1.2cm,] (A) \u003cspan class=\"nb\"\u003e{}\u003c/span\u003e (A.corner 1)\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in\u003cspan class=\"nb\"\u003e{\u003c/span\u003e1, 3, 5, 2, 4\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-- (A.corner \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e) \u003cspan class=\"nb\"\u003e}\u003c/span\u003e -- cycle;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe final result of our work will produce an image as seen at the top of this article. It should be noted that we are given the freedom to determine the position of the moon and stars as well as their coloring, for example I will change the position of the moon and stars and their coloring as shown in the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi13FE7q3Q9MZ7tLzDl9E-kBFLwE_MsWgzeo9K_HS201jAFrPzYJxNsRDg8r4S5kdQwlDLyoJm7SqWhbFa9Hbi-T4iPc5lb034MZrrKrv6ygnrtQuriQIDkFyG2K3M3lhQbXWqnYXU2SIg/s458/Screenshot+2021-06-11+21.58.10.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bulan dan bintang di latex\" border=\"0\" data-original-height=\"453\" data-original-width=\"458\" height=\"317\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi13FE7q3Q9MZ7tLzDl9E-kBFLwE_MsWgzeo9K_HS201jAFrPzYJxNsRDg8r4S5kdQwlDLyoJm7SqWhbFa9Hbi-T4iPc5lb034MZrrKrv6ygnrtQuriQIDkFyG2K3M3lhQbXWqnYXU2SIg/w320-h317/Screenshot+2021-06-11+21.58.10.png\" title=\"bulan dan bintang\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eSo, have you found the way? If not, please pay attention to the following command code.\u003c/p\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\documentclass{article}\\usepackage[x11names,dvipsnames]{xcolor}\\definecolor{bulanbintang}{rgb} {0,0.65,0.2}\\usepackage{tikz}\\usetikzlibrary{shapes.geometric}\\title{bulan bintang}\\author{aantriono82 }\\date{June 2021}\\begin{document}\\centering\\maketitle\\begin{tikzpicture}\\fill[bulanbintang,even odd rule] (0,0) circle (2.5);\\fill[white] (-.54,0) circle (2);\\fill[bulanbintang] (-2.5,0) node [regular polygon, regular polygon sides=5, inner sep=0pt, minimum size=1.2cm,] (A) {} (A.corner 1)\\foreach \\i in{1, 3, 5, 2, 4} {-- (A.corner \\i) } -- cycle;\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eAs seen above, the position of the moon is facing up, then facing to the left, and now we write the code for the image the moon faces right. For details, pay attention to the following code.\u003c/p\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\documentclass{article}\\usepackage[x11names,dvipsnames]{xcolor}\\definecolor{bulanbintang}{rgb} {0,65,2}\\usepackage{tikz}\\usetikzlibrary{shapes.geometric}\\title{bulan bintang}\\author{aantriono82 }\\date{June 2021}\\begin{document}\\centering\\maketitle\\begin{tikzpicture}\\fill[bulanbintang,even odd rule] (0,0) circle (2.5);\\fill[white] (.54,0) circle (2);\\fill[bulanbintang] (2.5,0) node [regular polygon, regular polygon sides=5, inner sep=0pt, minimum size=1.2cm,] (A) {} (A.corner 1)\\foreach \\i in{1, 3, 5, 2, 4} {-- (A.corner \\i) } -- cycle;\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003ePlease compile it and see the results, it turns out the image is facing to the right and the color has changed. In essence, when we change the position of the image, we only need to change the image coordinates listed in the 14th and 15th line commands. That\u0026rsquo;s the article about how to draw the moon and stars using LaTeX. If there are errors, please correct them and if you have input, please write them here. I hope this is helpful. Reference : LaTeXing Group\u003c/p\u003e", "language": "en" }, { "title" : "Drawing North Korean Flag", "date_published" : "2023-03-01T13:24:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-bendera-korea-utara/", "url" : "https://www.aantriono.com/en/menggambar-bendera-korea-utara/", "summary": " Drawing North Korean Flag in LaTeXAfter previously discussing Drawing Turkish Flag Using LaTeX\n, on this occasion the article is also still struggling about drawing flags. So, here I will provide a guide on how to draw the North Korean flag using LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-north-korean-flag-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-north-korean-flag-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing North Korean Flag in LaTeX\"\u003e\u003c/a\u003eDrawing North Korean Flag in LaTeX\u003c/h2\u003e\u003cp\u003eAfter previously discussing \u003ca href=\"https://www.aantriono.com/2021/06/menggambar-bendera-turki.html\" target=\"_blank\"\u003e\u003cstrong\u003eDrawing Turkish Flag Using LaTeX\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e, on this occasion the article is also still struggling about drawing flags. So, here I will provide a guide on how to draw the North Korean flag using LaTeX.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003cp\u003eWhy draw a flag again? Yup, because there are several similarities between the Turkish flag and the North Korean flag. For more details, look at the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMkblp57tNSjvtkeG9Dg5c7TEn33o8w2dQ3RQeCOOk4hTn9Nj-mNa39lgZ6BzINXcW-g4DKAMV4jClYQPT-RwbRfAUCt4xFzGmt30tt4itf3nhGwTPAD_0Rgv6Rgj5aGgOyjR_-U0NPrE/s607/Screenshot+2021-06-25+13.53.46.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"bendera korea utara menggunakan latex\" border=\"0\" data-original-height=\"376\" data-original-width=\"607\" height=\"198\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMkblp57tNSjvtkeG9Dg5c7TEn33o8w2dQ3RQeCOOk4hTn9Nj-mNa39lgZ6BzINXcW-g4DKAMV4jClYQPT-RwbRfAUCt4xFzGmt30tt4itf3nhGwTPAD_0Rgv6Rgj5aGgOyjR_-U0NPrE/w320-h198/Screenshot+2021-06-25+13.53.46.png\" title=\"bendera korea utara\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eIf we look at the picture of the flag, there are several parts that we will draw, namely: a dark blue rectangular box at the top, then a white rectangular box below it, then a red rectangular box, then a white circle, a red star right on the circle, a white rectangular box and finally a dark blue rectangular box.OK, let\u0026rsquo;s just discuss how to draw the North Korean flag using LaTeX.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003c!--more--\u003e\n\u003ch2 id=\"step-by-step-tikz-construction\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-tikz-construction\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step TikZ Construction\"\u003e\u003c/a\u003eStep-by-Step TikZ Construction\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen a text editor.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSelect the article document class using the following command.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\documentclass{article}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eIn the preamble section, determine the packages that will be used, namely via the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\usepackage[x11names,dvipsnames]{xcolor}\\definecolor{bulanbintang}{rgb} {255,255,255}\\definecolor{cobalt}{rgb}{0.0, 0.28, 0.67}\\usepackage{tikz}\\usetikzlibrary{shapes.geometric}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eWrite the title, author, and date of creation of the document using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\title{Bendera Korea Utara}\\author{Your Name }\\date{\\today}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eNext, we enter the body of the document, and write the following command to display the document title, author, and date the document was created.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\begin{document}\\centering\\maketitle\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eIt\u0026rsquo;s time to write down the drawing command as follows.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\begin{tikzpicture}\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eFirst, we draw a dark blue rectangular box using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[cobalt] (0, 4) rectangle (15, 5);\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eThen draw a white rectangular box using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[white] (0, 3.7) rectangle (15, 4);\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"9\"\u003e\n\u003cli\u003eThen draw a red rectangular box using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill [red] (0,-2.5) rectangle (15,3.7);\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"10\"\u003e\n\u003cli\u003eThen draw a white circle using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[bulanbintang,even odd rule] (5,.5) circle (2.3);\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"11\"\u003e\n\u003cli\u003eThen draw a star using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[bulanbintang, red] (5,0.5) node [ regular polygon, regular polygon sides=5, inner sep=0pt, minimum size=4.5cm] (A) {} (A.corner 1)\\foreach \\i in{1, 3, 5, 2, 4} {-- (A.corner \\i) } -- cycle;\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"12\"\u003e\n\u003cli\u003eThen draw a white rectangular box using the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[white] (0, -2.8) rectangle (15, -2.5);\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"13\"\u003e\n\u003cli\u003eLastly we draw a dark blue rectangular box via the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\fill[cobalt] (0, -3.8) rectangle (15, -2.8);\u003c/code\u003e\u003c/pre\u003e\n\u003col start=\"14\"\u003e\n\u003cli\u003eThe final step is that we give the command to end the image and close the document via the following command.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThe final result of the document after compiling looks like the first image in this article. How are you, sir? Have you put it into practice and succeeded, or have you not succeeded and there are errors? If there is an error, please try again and pay attention to the instructions. So that you understand more, Below is a complete guide.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"#\" style=\"border: 1px solid rgb(204, 204, 204); padding: 3px 7px; text-decoration: none;\"\u003e+ Show/Hide\u003c/a\u003e\u003c/p\u003e\n\u003cdiv id=\"toggle\" style=\"background: rgb(255, 255, 255); display: none; padding: 10px;\"\u003e*\n\u003cpre\u003e\u003ccode class=\"css\"\u003e\\documentclass{article}\\usepackage{fancybox}\\usepackage[x11names,dvipsnames]{xcolor}\\definecolor{bulanbintang}{rgb} {255,255,255}\\definecolor{cobalt}{rgb}{0.0, 0.28, 0.67}\\usepackage{tikz}\\usetikzlibrary{shapes.geometric}\\title{Bendera Korea Utara}\\author{Aan Triono }\\date{\\today}\\begin{document}\\centering\\maketitle\\begin{center}\\Large \\textcolor{blue}{\\ovalbox{Menggambar Bendera Korea Utara Menggunakan \\LaTeX}}\\end{center}\\vspace{1cm}\\begin{tikzpicture}\\fill[cobalt] (0, 4) rectangle (15, 5);\\fill[white] (0, 3.7) rectangle (15, 4);\\fill [red] (0,-2.5) rectangle (15,3.7);\\fill[bulanbintang,even odd rule] (5,.5) circle (2.3);\\fill[bulanbintang, red] (5,0.5) node [ regular polygon, regular polygon sides=5, inner sep=0pt, minimum size=4.5cm] (A) {} (A.corner 1)\\foreach \\i in{1, 3, 5, 2, 4} {-- (A.corner \\i) } -- cycle;\\fill[white] (0, -2.8) rectangle (15, -2.5);\\fill[cobalt] (0, -3.8) rectangle (15, -2.8);\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n* This is a guide on how to draw the North Korean flag using LaTeX.\n\u003c!--more--\u003e\n\u003c!--more--\u003e\n\u003cp\u003eIf there are errors, please correct them, if you have any input, please write them here. I hope this is helpful.\u003c/p\u003e", "language": "en" }, { "title" : "How to Draw a Cup of Coffee Using LaTeX", "date_published" : "2023-03-01T13:20:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-menggambar-secangkir-kopi/", "url" : "https://www.aantriono.com/en/cara-menggambar-secangkir-kopi/", "summary": " IntroductionIn this article, I will share an article about how to draw a cup of coffee using LaTeX\\LaTeXLATE​X. Yup\u0026hellip; a cup of coffee is usually served and enjoyed in the morning to start activities, in the afternoon when resting, or even served in the evening to accompany staying up late, heeee. OK, let\u0026rsquo;s go straight to the tutorial on how to draw a cup of coffee using the word processor LaTeX\\LaTeXLATE​X.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn this article, I will share an article about how to draw a cup of coffee using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Yup\u0026hellip; a cup of coffee is usually served and enjoyed in the morning to start activities, in the afternoon when resting, or even served in the evening to accompany staying up late, heeee. OK, let\u0026rsquo;s go straight to the tutorial on how to draw a cup of coffee using the word processor \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv class=\"bacajuga\"\u003e\n\u003ch4 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also Read\"\u003e\u003c/a\u003eAlso Read\u003c/h4\u003e\u003cul\u003e \u003cli\u003e\u003ca href=\"/2021/06/menggambar-bendera-turki.html\"\u003eDrawing Turkish Flag\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"/2021/06/menggambar-bendera-korea-utara.html\"\u003eDrawing North Korean Flag\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"/2021/04/cara-menggambar-unsur-unsur-lingkaran.html\"\u003eDrawing Circle Elements\u003c/a\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/div\u003e\n\u003ch2 id=\"preamble\" class=\"headerLink\"\u003e\n \u003ca href=\"#preamble\" class=\"header-mark\" aria-label=\"Direct link to section: Preamble\"\u003e\u003c/a\u003ePreamble\u003c/h2\u003e\u003cp\u003eIn the preamble, we type commands about the document class and packages needed to draw the cup of coffee.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\documentclass{article}\\usepackage{tikz}\\usetikzlibrary{fadings}\\tikzfading[name=fade out, inner color=transparent!0, outer color=transparent!100]\u003c/code\u003e\u003c/pre\u003e\n\u003ch2 id=\"document-body\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-body\" class=\"header-mark\" aria-label=\"Direct link to section: Document Body\"\u003e\u003c/a\u003eDocument Body\u003c/h2\u003e\u003cp\u003eIn the body of the document, we will draw the parts consisting of the cup base, cup, cup handle, and coffee water. Here all parts of the image are light blue.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{document}\\begin{tikzpicture} \\foreach \\c [count=\\i from 0] in {blue!25}{ \\tikzset{xshift={mod(\\i,2)*3cm}, yshift=-floor(\\i/2)*3cm} \\colorlet{cup}{\\c} }\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"cup-base\" class=\"headerLink\"\u003e\n \u003ca href=\"#cup-base\" class=\"header-mark\" aria-label=\"Direct link to section: Cup base\"\u003e\u003c/a\u003eCup base\u003c/h3\u003e\u003cp\u003eFirst, we draw a cup base shaped like a circle with a depression in the center for the cup.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{scope}[shift={(0,-1-1/16)}] \\fill [black!87.5, path fading=fade out] (0,-2/8) ellipse [x radius=6/4, y radius=3/4]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3}] (0,0) ++(180:5/4) arc (180:360:5/4 and 5/8+1/16); \\fill [cup, postaction={left color=black!50, right color=white, opacity=1/3}] (0,0) ellipse [x radius=5/4, y radius=5/8]; \\fill [cup, postaction={left color=white, right color=black, opacity=1/3}] (0,1/16) ellipse [x radius=5/4/2, y radius=5/8/2]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3}] (0,0) ellipse [x radius=5/4/2-1/16, y radius=5/8/2-1/16]; \\end{scope} \u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"cup\" class=\"headerLink\"\u003e\n \u003ca href=\"#cup\" class=\"header-mark\" aria-label=\"Direct link to section: Cup\"\u003e\u003c/a\u003eCup\u003c/h3\u003e\u003cp\u003eThe cup is placed right in the middle of the cup base so that it covers the hollow area in the cup base.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\fill [cup!25!black, path fading=fade out] (0,-1-1/16) ellipse [x radius=3/4, y radius=1/3]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3/2}, postaction={bottom color=black, top color=white, opacity=1/3/2}] (-1,0) arc (180:360:1 and 5/4); \\fill [cup, postaction={left color=white, right color=black, opacity=1/3}] (0,0) ellipse [x radius=1, y radius=1/2]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3/2}, postaction={bottom color=black, top color=white, opacity=1/3/2}] (0,0) ellipse [x radius=1-1/16, y radius=1/2-1/16];\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"cup-handle\" class=\"headerLink\"\u003e\n \u003ca href=\"#cup-handle\" class=\"header-mark\" aria-label=\"Direct link to section: Cup handle\"\u003e\u003c/a\u003eCup handle\u003c/h3\u003e\u003cp\u003eThe cup handle is located to the right of the cup, starting from the center to the top of the cup.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e \\begin{scope}[shift=(10:7/8), rotate=-30, yslant=1/2, xslant=-1/8] \\fill [cup, postaction={top color=black, bottom color=white, opacity=1/3}] (0,0) arc (130:-100:3/8 and 1/2) -- ++(0,1/4) arc (-100:130:1/8 and 1/4) -- cycle; \\fill [cup, postaction={top color=white, bottom color=black, opacity=1/3}] (0,0) arc (130:-100:3/8 and 1/2) -- ++(0,1/32) arc (-100:130:1/4 and 1/3) -- cycle; \\end{scope}\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"coffee-water\" class=\"headerLink\"\u003e\n \u003ca href=\"#coffee-water\" class=\"header-mark\" aria-label=\"Direct link to section: Coffee water\"\u003e\u003c/a\u003eCoffee water\u003c/h3\u003e\u003cp\u003eThe color of the coffee water here is a combination of black and brown, with a color composition adjusted to our wishes.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e \\begin{scope} \\clip ellipse [x radius=1-1/16, y radius=1/2-1/16]; \\fill [brown!10!black] (0,-1/4) ellipse [x radius=3/4, y radius=3/8]; \\fill [brown!40!black, path fading=fade out] (0,-1/4) ellipse [x radius=3/4, y radius=3/8]; \\end{scope}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eYou can change the color of the coffee water by adjusting the coloration level \u003cem\u003ebrown\u003c/em\u003e And *black. *\u003c/p\u003e\n\u003ch2 id=\"complete-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete Code\"\u003e\u003c/a\u003eComplete Code\u003c/h2\u003e\u003cp\u003eIf you want to try compiling a coffee cup drawing document script using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, please copy the following code.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\documentclass{article}\\usepackage{tikz}\\usetikzlibrary{fadings}\\tikzfading[name=fade out, inner color=transparent!0, outer color=transparent!100]\\begin{document}\\begin{tikzpicture} \\foreach \\c [count=\\i from 0] in {blue!25}{ \\tikzset{xshift={mod(\\i,2)*3cm}, yshift=-floor(\\i/2)*3cm} \\colorlet{cup}{\\c} % ~~~~~~~~~ Menggambar alas cangkir ~~~~~~~~~~~~~ \\begin{scope}[shift={(0,-1-1/16)}] \\fill [blue!87.5, path fading=fade out] (0,-2/8) ellipse [x radius=6/4, y radius=3/4]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3}] (0,0) ++(180:5/4) arc (180:360:5/4 and 5/8+1/16); \\fill [cup, postaction={left color=black!50, right color=white, opacity=1/3}] (0,0) ellipse [x radius=5/4, y radius=5/8]; \\fill [cup, postaction={left color=white, right color=black, opacity=1/3}] (0,1/16) ellipse [x radius=5/4/2, y radius=5/8/2]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3}] (0,0) ellipse [x radius=5/4/2-1/16, y radius=5/8/2-1/16]; \\end{scope} % ~~~~~~~~~~~~ Menggambar gagang cangkir ~~~~~~~~~~~~~~~~~~ \\begin{scope}[shift=(10:7/8), rotate=-30, yslant=1/2, xslant=-1/8] \\fill [cup, postaction={top color=black, bottom color=white, opacity=1/3}] (0,0) arc (130:-100:3/8 and 1/2) -- ++(0,1/4) arc (-100:130:1/8 and 1/4) -- cycle; \\fill [cup, postaction={top color=white, bottom color=black, opacity=1/3}] (0,0) arc (130:-100:3/8 and 1/2) -- ++(0,1/32) arc (-100:130:1/4 and 1/3) -- cycle; \\end{scope} % ~~~~~~~~ Menggambar cangkir ~~~~~~~~~~~~~~ \\fill [cup!25!black, path fading=fade out] (0,-1-1/16) ellipse [x radius=3/4, y radius=1/3]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3/2}, postaction={bottom color=black, top color=white, opacity=1/3/2}] (-1,0) arc (180:360:1 and 5/4); \\fill [cup, postaction={left color=white, right color=black, opacity=1/3}] (0,0) ellipse [x radius=1, y radius=1/2]; \\fill [cup, postaction={left color=black, right color=white, opacity=1/3/2}, postaction={bottom color=black, top color=white, opacity=1/3/2}] (0,0) ellipse [x radius=1-1/16, y radius=1/2-1/16]; % ~~~~~~~~~~~~ Menggambar air kopi ~~~~~~~~~~~~~~~ \\begin{scope} \\clip ellipse [x radius=1-1/16, y radius=1/2-1/16]; \\fill [brown!10!black] (0,-1/4) ellipse [x radius=3/4, y radius=3/8]; \\fill [brown!40!black, path fading=fade out] (0,-1/4) ellipse [x radius=3/4, y radius=3/8]; \\end{scope} }\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eNext, input the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e code into the LaTeX engine on this blog in the LaTeX Engine tools \u003ca href=\"https://www.silagebra.com/p/latex-engine.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e.The compiled results of writing these commands in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e will look like the following image. \u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSsZbqWhMvZvHZKojlJzdAs6bM5ulf3lJy9HscppEpAjhAF9kSYWysJsXWNLx-bTbiIU39jzsqjoxqrqjyZoxWuVr8etA1nwMxoFhDFKNfqJrQ2tXjAFdg_AgVO1kAz3k-ftrKVWSDAfE/s450/kopi.PNG\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"secangkir kopi di LaTeX\" border=\"0\" data-original-height=\"377\" data-original-width=\"450\" height=\"268\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSsZbqWhMvZvHZKojlJzdAs6bM5ulf3lJy9HscppEpAjhAF9kSYWysJsXWNLx-bTbiIU39jzsqjoxqrqjyZoxWuVr8etA1nwMxoFhDFKNfqJrQ2tXjAFdg_AgVO1kAz3k-ftrKVWSDAfE/w320-h268/kopi.PNG\" title=\"secangkir kopi\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThese are the steps for drawing a cup of coffee using LaTeX. Of course you can change and adjust the shape or coloring. I hope this is helpful.\u003cem\u003eReference: texample.net\u003c/em\u003e\u003c/p\u003e\n", "language": "en" }, { "title" : "Skydiving Parachute", "date_published" : "2023-03-01T13:20:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/parasut-terjun-payung/", "url" : "https://www.aantriono.com/en/parasut-terjun-payung/", "summary": " Drawing a Skydiving Parachute in LaTeXIn this tutorial we will draw a skydiving parachute in LaTeX. The illustration has three main parts:\nthe canopy made from circular arcs the suspension lines the small owl hanging below The final result looks like this.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"drawing-a-skydiving-parachute-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-a-skydiving-parachute-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing a Skydiving Parachute in LaTeX\"\u003e\u003c/a\u003eDrawing a Skydiving Parachute in LaTeX\u003c/h2\u003e\u003cp\u003eIn this tutorial we will draw a skydiving parachute in LaTeX. The illustration has three main parts:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe canopy made from circular arcs\u003c/li\u003e\n\u003cli\u003ethe suspension lines\u003c/li\u003e\n\u003cli\u003ethe small owl hanging below\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe final result looks like this.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfAuJjUusnGHJoHwveg_FuuxM6szSqx1SgYI3ksy12SJ9yg-7ctsyvAZAwwSVnQ8MgT5SMMadPWqRGbocE7KBizI__N3oKlMxp4Z0vukE1ZSz0qVVEDnS7WqnydTniiMNJpgTOkBm3MyE/s582/Screenshot+2021-06-15+13.47.37.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"skydiving parachute in latex\" border=\"0\" data-original-height=\"582\" data-original-width=\"505\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfAuJjUusnGHJoHwveg_FuuxM6szSqx1SgYI3ksy12SJ9yg-7ctsyvAZAwwSVnQ8MgT5SMMadPWqRGbocE7KBizI__N3oKlMxp4Z0vukE1ZSz0qVVEDnS7WqnydTniiMNJpgTOkBm3MyE/w278-h320/Screenshot+2021-06-15+13.47.37.png\" title=\"skydiving parachute\" width=\"278\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"step-by-step-tikz-construction\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-tikz-construction\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step TikZ Construction\"\u003e\u003c/a\u003eStep-by-Step TikZ Construction\u003c/h2\u003e\u003cp\u003eFollow these steps.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eOpen your LaTeX editor.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eUse the \u003ccode\u003estandalone\u003c/code\u003e document class with a \u003ccode\u003e1cm\u003c/code\u003e border.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - 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https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\documentclass\u003c/span\u003e\u003cspan class=\"na\"\u003e[border=1cm]\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003estandalone\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eLoad the \u003ccode\u003etikz\u003c/code\u003e package.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-2\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikz\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eTo draw the owl, load \u003ccode\u003etikzlings-owls\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\usepackage\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzlings-owls\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eStart the document body and the drawing environment.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eDraw the main canopy as a semicircle.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0:3.5) arc (0:180:3.5);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe first result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiC5-BmeBnGy426vVSPE10R2HgDGg1HnO_nRkf7rKIlvk7bDEnLPNKFAS9B5V1iSvU3-XZOcb1WoUCLVmPKGTsxw-Fe2Lxwm1Le8G1R2rwGlbZjLHFeKeAg53Pgx7tGw8DVCV5k67tvW9U/s1231/Screenshot+2021-06-15+11.48.59.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"parachute canopy in latex\" border=\"0\" data-original-height=\"316\" data-original-width=\"1231\" height=\"82\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiC5-BmeBnGy426vVSPE10R2HgDGg1HnO_nRkf7rKIlvk7bDEnLPNKFAS9B5V1iSvU3-XZOcb1WoUCLVmPKGTsxw-Fe2Lxwm1Le8G1R2rwGlbZjLHFeKeAg53Pgx7tGw8DVCV5k67tvW9U/w320-h82/Screenshot+2021-06-15+11.48.59.png\" title=\"parachute canopy\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eAdd the smaller arcs under the canopy to form the attachment points for the lines.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-6\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (-3.5,0) arc (180:0:0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (-2.5,0) arc (180:0:0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (-1.5,0) arc (180:0:0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (-0.5,0) arc (180:0:0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0.5,0) arc (180:0:0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (1.5,0) arc (180:0:0.5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (2.5,0) arc (180:0:0.5);\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjteQL-qo0j2E7qSOUw9jKUWa-kKX1AtyV9Qt_WYqswvRTRAVoF1H3VP2HVtzRF85rNL5W1ucZoZ0PDIpyXnzcpuxXFVTBRXHtMvINhwsrapXrM0-_Nzp0MAXu_0NpOBWS8AFanFhiKw6I/s1159/Screenshot+2021-06-15+11.52.49.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"parachute canopy in latex\" border=\"0\" data-original-height=\"356\" data-original-width=\"1159\" height=\"98\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjteQL-qo0j2E7qSOUw9jKUWa-kKX1AtyV9Qt_WYqswvRTRAVoF1H3VP2HVtzRF85rNL5W1ucZoZ0PDIpyXnzcpuxXFVTBRXHtMvINhwsrapXrM0-_Nzp0MAXu_0NpOBWS8AFanFhiKw6I/w320-h98/Screenshot+2021-06-15+11.52.49.png\" title=\"parachute canopy\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eSteps 6 and 7 can be shortened into a single command:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-7\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (0:3.5) arc (0:180:3.5)\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e1,...,7\u003cspan class=\"nb\"\u003e}{\u003c/span\u003earc (180:0:0.5)\u003cspan class=\"nb\"\u003e}\u003c/span\u003e -- cycle;\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eTo color the canopy, add these options after \u003ccode\u003e\\draw\u003c/code\u003e:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-8\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e[left color=red!80,right color=green!80]\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe colored canopy looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-osoOqzuWNKPd8aIH57rw-URrB-FoiqBDTsgulfDbLky8M1DcKy3GlV8_Mnj-tP3pMJQjD_bRYZCV_rgVMXLnwsH4YWF1vjmdUIYFbYHoxYCwwhpV6pCEym2YrV9zAsp2wUi4w13bkvI/s501/Screenshot+2021-06-15+13.20.04.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"parachute canopy in latex\" border=\"0\" data-original-height=\"304\" data-original-width=\"501\" height=\"194\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-osoOqzuWNKPd8aIH57rw-URrB-FoiqBDTsgulfDbLky8M1DcKy3GlV8_Mnj-tP3pMJQjD_bRYZCV_rgVMXLnwsH4YWF1vjmdUIYFbYHoxYCwwhpV6pCEym2YrV9zAsp2wUi4w13bkvI/w320-h194/Screenshot+2021-06-15+13.20.04.png\" title=\"parachute canopy\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eDraw the suspension lines.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-9\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\foreach\u003c/span\u003e \u003cspan class=\"k\"\u003e\\i\u003c/span\u003e in \u003cspan class=\"nb\"\u003e{\u003c/span\u003e-3.5,...,3.5\u003cspan class=\"nb\"\u003e}\u003c/span\u003e \u003cspan class=\"nb\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\draw\u003c/span\u003e (\u003cspan class=\"k\"\u003e\\i\u003c/span\u003e,0) -- (0,-5);\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMSzSmFUYEnQyq59jLds2KBhYFramUtk4ZZqnMK66ix9VijYiE6La4-voYaWb8NdcjhyphenhyphenpCb8eDsMSveL5uuNQKfj9mvgG0f4JEaHbCpqr-j5OIml6h3EbiMzngvtIMKde8wZCetXgYtUg/s554/Screenshot+2021-06-15+13.25.08.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"skydiving parachute in latex\" border=\"0\" data-original-height=\"554\" data-original-width=\"501\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMSzSmFUYEnQyq59jLds2KBhYFramUtk4ZZqnMK66ix9VijYiE6La4-voYaWb8NdcjhyphenhyphenpCb8eDsMSveL5uuNQKfj9mvgG0f4JEaHbCpqr-j5OIml6h3EbiMzngvtIMKde8wZCetXgYtUg/w289-h320/Screenshot+2021-06-15+13.25.08.png\" title=\"skydiving parachute\" width=\"289\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"9\"\u003e\n\u003cli\u003ePlace the owl below the lines.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-10\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e[yshift=-6.5cm]\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"k\"\u003e\\owl\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003escope\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe result looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZDMQa5n6VQh8tILwYtvly5tkfq_pwH_-TU2t02Dhdf-2KEfT1wiRqd0U1gmSNgdoQ-VidMXFWnjNiGvmJ6fcLREoHumiPHHrY7KVqtHD8L_ySdZ02miePeRwFeZzQxwnY3JS0OWlH_mM/s554/Screenshot+2021-06-15+13.39.59.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"skydiving parachute in latex\" border=\"0\" data-original-height=\"554\" data-original-width=\"429\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZDMQa5n6VQh8tILwYtvly5tkfq_pwH_-TU2t02Dhdf-2KEfT1wiRqd0U1gmSNgdoQ-VidMXFWnjNiGvmJ6fcLREoHumiPHHrY7KVqtHD8L_ySdZ02miePeRwFeZzQxwnY3JS0OWlH_mM/w248-h320/Screenshot+2021-06-15+13.39.59.png\" title=\"skydiving parachute\" width=\"248\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"10\"\u003e\n\u003cli\u003eTo make the owl look more playful, add a top hat.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-11\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\thing\u003c/span\u003e\u003cspan class=\"na\"\u003e[tophat,scale=1.5,yshift=-0.6cm,xshift=-0.05cm]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"11\"\u003e\n\u003cli\u003eAdd a speech thought bubble as well.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-12\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\thing\u003c/span\u003e\u003cspan class=\"na\"\u003e[think={\\small How are you?},scale=1.5,yshift=-0.6cm,xshift=0.3cm]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003col start=\"12\"\u003e\n\u003cli\u003eClose the drawing and the document.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-13\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003etikzpicture\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003edocument\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eThe final result after compilation looks like this.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh31Zkusgv8rpn7z8YPuIMOCOdc9rwGtS_hal6c1H4ICQ16vw-oisHyyW7WUX_xKKFKtS4OJrRgIOPVACd0XMkMlcxKkY_Nhq6vIg9jXdPi_yhmDtNNp68ya_FNoepB0df2aeoeGaBhj2I/s582/Screenshot+2021-06-15+13.47.37.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"skydiving parachute in latex\" border=\"0\" data-original-height=\"582\" data-original-width=\"505\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh31Zkusgv8rpn7z8YPuIMOCOdc9rwGtS_hal6c1H4ICQ16vw-oisHyyW7WUX_xKKFKtS4OJrRgIOPVACd0XMkMlcxKkY_Nhq6vIg9jXdPi_yhmDtNNp68ya_FNoepB0df2aeoeGaBhj2I/w278-h320/Screenshot+2021-06-15+13.47.37.png\" title=\"skydiving parachute\" width=\"278\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThat is a simple step-by-step guide to drawing a skydiving parachute in LaTeX. For more TikZ drawing inspiration, you can also browse \u003ca href=\"https://latexdraw.com/\" target=\"_blank\"\u003elatexdraw.com\u003c/a\u003e.\u003c/p\u003e", "language": "en" }, { "title" : "Drawing Mitsubishi Logo", "date_published" : "2023-03-01T13:18:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-logo-mitsubishi/", "url" : "https://www.aantriono.com/en/menggambar-logo-mitsubishi/", "summary": " IntroductionWhen it was founded in 1870 AD, Mitsubishi was a shipping company. The founder was Yataro Iwasaki (1834-1885). In 1873 the company changed its name to Mitsubishi Shokai. The name Mitsubishi itself is composed of two words, namely Mitsu And hishi which turned into bushi. Mitsu means* three* and bishi means under my lace. Therefore, the Mitsubishi logo is three rhombuses or three diamonds. So, on this occasion, I will not buy a Mitsubishi Xpander or Pajero, heeeee. But just sharing a tutorial on how to draw the Mitsubishi logo using LaTeX\\LaTeXLATE​X in polar coordinates and in Cartesian coordinates 😂.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eWhen it was founded in 1870 AD, Mitsubishi was a shipping company. The founder was Yataro Iwasaki (1834-1885). In 1873 the company changed its name to Mitsubishi Shokai. The name Mitsubishi itself is composed of two words, namely \u003cem\u003eMitsu\u003c/em\u003e And \u003cem\u003ehishi\u003c/em\u003e which turned into \u003cem\u003ebus\u003c/em\u003ehi. Mitsu means* three* and bishi means \u003cem\u003eunder my lace\u003c/em\u003e. Therefore, the Mitsubishi logo is \u003cem\u003ethree rhombuses\u003c/em\u003e or \u003cem\u003ethree diamonds.\u003c/em\u003e So, on this occasion, I will not buy a Mitsubishi Xpander or Pajero, heeeee. But just sharing a tutorial on how to draw the Mitsubishi logo using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e in polar coordinates and in Cartesian coordinates 😂.\u003c/p\u003e\n\u003ch2 id=\"drawing-the-mitsubishi-logo-in-polar-coordinates\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-the-mitsubishi-logo-in-polar-coordinates\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing the Mitsubishi Logo in Polar Coordinates\"\u003e\u003c/a\u003eDrawing the Mitsubishi Logo in Polar Coordinates\u003c/h2\u003e\u003cp\u003eFor the first Mitsubishi logo, we will use polar coordinates. The steps are as follows. PreambleIn the preamble section we determine the class of documents and packages required. For example as follows.\\documentclass[border=0.2cm]{article}\\usepackage{tikz}Document BodyIn the body of the document we start drawing using the tikz package commands starting with \\begin{tikzpicture} and ending with \\end{tikzpicture}. And the Mitsubishi logo that will be drawn is three diamonds. To make it easier we call them top diamonds, right diamonds, and left diamonds. Top Diamond% ~~~~~~~~~~~~~~ Berlian Atas ~~~~~~~~~~~~~~~~~\\fill[red] (0,0) \u0026ndash; (60:2) \u0026ndash; (90:3.5) \u0026ndash; (120:2) \u0026ndash; (0,0);Right Diamond% ~~~~~~~~~~~~~ Berlian Kanan ~~~~~~~~~~~~~~~~~\\fill[red] (0,0) \u0026ndash; (0:2) \u0026ndash; (-30:3.5) \u0026ndash; (-60:2) \u0026ndash; (0,0);Left Diamond% ~~~~~~~~~~~~~~ Berlian Kiri ~~~~~~~~~~~~~~~~~ \\fill[red] (0,0) \u0026ndash; (180:2) \u0026ndash; (210:3.5) \u0026ndash; (240:2) \u0026ndash; (0,0);After all the code has been written and the document has been given a closing code, the manuscript is then compiled. The compilation results look like the following image. Changing ColorsIf you want to change the coloring of each diamond, or divide each diamond into two different colored parts, you can try the following code. Pay attention to the color codes written in the following command codes. You can use it or change it to another color. Top Diamond% ~~~~~~~~~~~~~ Berlian atas ~~~~~~~~~~~~~~~\\fill[magenta] (0,0) \u0026ndash; (60:2) \u0026ndash; (90:3.5);\\fill[teal] (0,0) \u0026ndash; (120:2) \u0026ndash; (90:3.5);Right Diamond% ~~~~~~~~~~~~ Berlian kanan ~~~~~~~~~~~~~~~ \\fill[red] (0,0) \u0026ndash; (0:2) \u0026ndash; (-30:3.5); \\fill[cyan] (0,0) \u0026ndash; (-60:2) \u0026ndash; (-30:3.5);Left Diamond% ~~~~~~~~~~~~~ Berlian Kiri ~~~~~~~~~~~~~~~ \\fill[blue] (0,0) \u0026ndash; (180:2) \u0026ndash; (210:3.5); \\fill[green] (0,0) \u0026ndash; (240:2) \u0026ndash; (30:-3.5);And the compilation results will look like the following image. Drawing the Mitsubishi Logo in Cartesian CoordinatesNow we will draw the Mitsubishi logo in Cartesian Coordinates. How? Follow the following guide. PreambleIn the preamble section we determine the class of documents and packages required. For example as follows.\\documentclass{article}\\usepackage[landscape, margin=1 cm]{geometry}\\usepackage{tikz}Document BodyIn the body of the document we write the command to start drawing and determine the coordinates of each corner of the diamond. For example as follows.\\begin{document}\\begin{tikzpicture}[thick,scale=5]% ~~~~~~~~~~~ Menggambar titik koordinat ~~~~~~~~~ \\coordinate (A) at (0, 0); \\coordinate (B) at (1, 0); \\coordinate (C) at (1.7, -0.9); \\coordinate (D) at (0.6, -0.9); \\coordinate (E) at (-1, 0); \\coordinate (F) at (-1.7, -0.9); \\coordinate (G) at (-0.6, -0.9); \\coordinate (H) at (0.5, 0.85); \\coordinate (I) at (0, 1.7); \\coordinate (J) at (-0.5, 0.85); Next, we will connect the coordinate points and color them to form three diamonds using the following command.%\u0026mdash;\u0026mdash;\u0026mdash;\u0026mdash;- Menggambar Logo Mitsubishi \u0026mdash;\u0026mdash;\u0026mdash;\u0026ndash; \\fill[red, very thick] (A) \u0026ndash; (B)\u0026ndash;(C)\u0026ndash;(D)\u0026ndash;(A); \\fill[red, very thick] (A) \u0026ndash; (E)\u0026ndash;(F)\u0026ndash;(G)\u0026ndash;(A); \\fill[red, very thick] (A) \u0026ndash; (H)\u0026ndash;(I)\u0026ndash;(J)\u0026ndash;(A);\\end{tikzpicture}\\end{document}After all the code has been written and the document has been given a closing code, the manuscript is then compiled. The compilation results look like the following image. Complete CodeBelow is the complete code for drawing the Mitsubishi logo in polar coordinates and Cartesian coordinates. Polar Coordinates\\documentclass[border=0.2cm]{article}\\usepackage{tikz}\\title{Logo Mitsubishi}\\author{Aan Triono}\\date{\\today}\\begin{document}\\maketitle\\centering\\begin{tikzpicture}% ~~~~~~~~~~~~~~ Berlian Atas ~~~~~~~~~~~~~~~~~\\fill[red] (0,0) \u0026ndash; (60:2) \u0026ndash; (90:3.5) \u0026ndash; (120:2) \u0026ndash; (0,0);% ~~~~~~~~~~~~~ Berlian Kanan ~~~~~~~~~~~~~~~~~ \\fill[red] (0,0) \u0026ndash; (0:2) \u0026ndash; (-30:3.5) \u0026ndash; (-60:2) \u0026ndash; (0,0);% ~~~~~~~~~~~~~~ Berlian Kiri ~~~~~~~~~~~~~~~~~ \\fill[red] (0,0) \u0026ndash; (180:2) \u0026ndash; (210:3.5) \u0026ndash; (240:2) \u0026ndash; (0,0); \t \\end{tikzpicture}\\end{document}Different Color Polar Coordinates\\documentclass[border=0.2cm]{article}\\usepackage{tikz}\\title{Logo Mitsubishi}\\author{Aan Triono}\\date{\\today}\\begin{document}\\maketitle\\centering\\begin{tikzpicture}% ~~~~~~~~~~~~~ Berlian atas ~~~~~~~~~~~~~~~\\fill[magenta] (0,0) \u0026ndash; (60:2) \u0026ndash; (90:3.5);\\fill[teal] (0,0) \u0026ndash; (120:2) \u0026ndash; (90:3.5);% ~~~~~~~~~~~~~ Berlian Kiri ~~~~~~~~~~~~~~~ \\fill[blue] (0,0) \u0026ndash; (0:-2) \u0026ndash; (210:3.5); \\fill[green] (0,0) \u0026ndash; (240:2) \u0026ndash; (30:-3.5);% ~~~~~~~~~~~~ Berlian kanan ~~~~~~~~~~~~~~~ \\fill[red] (0,0) \u0026ndash; (0:2) \u0026ndash; (-30:3.5); \\fill[cyan] (0,0) \u0026ndash; (-60:2) \u0026ndash; (-30:3.5); \t \\end{tikzpicture}\\end{document}Cartesian coordinatescumentclass{article}\\usepackage[landscape, margin=1 cm]{geometry}\\usepackage{tikz}\\title{Logo Mitsubishi}\\author{Aan Triono}\\date{\\today}\\begin{document}\\maketitle\\centering\\vspace{.3 cm}\\begin{tikzpicture}[thick,scale=5]% ~~~~~~~~~~~ Menggambar titik koordinat ~~~~~~~~~ \\coordinate (A) at (0, 0); \\coordinate (B) at (1, 0); \\coordinate (C) at (1.7, -0.9); \\coordinate (D) at (0.6, -0.9); \\coordinate (E) at (-1, 0); \\coordinate (F) at (-1.7, -0.9); \\coordinate (G) at (-0.6, -0.9); \\coordinate (H) at (0.5, 0.85); \\coordinate (I) at (0, 1.7); \\coordinate (J) at (-0.5, 0.85); %\u0026mdash;\u0026mdash;\u0026mdash;\u0026mdash;- Menggambar Logo Mitsubishi \u0026mdash;\u0026mdash;\u0026mdash;\u0026ndash; \\fill[red, very thick] (A) \u0026ndash; (B)\u0026ndash;(C)\u0026ndash;(D)\u0026ndash;(A); \\fill[red, very thick] (A) \u0026ndash; (E)\u0026ndash;(F)\u0026ndash;(G)\u0026ndash;(A); \\fill[red, very thick] (A) \u0026ndash; (H)\u0026ndash;(I)\u0026ndash;(J)\u0026ndash;(A); \\end{tikzpicture}\\end{document}You can try writing it using the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e engine available on this blog and immediately see the results. How to? Copy all the code above and paste it here. Closing\u003c/p\u003e\n\u003cp\u003eThese are the steps for drawing the Mitsubishi logo using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e both in polar coordinates and in Cartesian coordinates. If there are errors, please correct them, and if you have input, please write them here. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Drawing a Logo Using LaTeX", "date_published" : "2023-03-01T10:50:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-logo-menggunakan-latex/", "url" : "https://www.aantriono.com/en/menggambar-logo-menggunakan-latex/", "summary": " IntroductionLogos are often used by a body or institution to introduce his identity. Logos usually take the form of a particular image or sketch which has its own meaning for the body or institution that uses it. Generally, to create logos, Adobe Photoshop, Corel Draw or applications that provide logo templates such as Canva are used. What about LaTeX\\LaTeXLATE​X, can it be used to draw a logo? Of course, with the PGFPlot or TikZ packages already available in LaTeX\\LaTeXLATE​X, we can draw a logo. Whether a logo is good or not is certainly influenced by many factors, one of which is the creativity of the creator. Now we will discuss the steps in drawing a logo using LaTeX\\LaTeXLATE​X.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eLogos are often used by a body or institution to introduce his identity. Logos usually take the form of a particular image or sketch which has its own meaning for the body or institution that uses it. Generally, to create logos, Adobe Photoshop, Corel Draw or applications that provide logo templates such as Canva are used. What about \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, can it be used to draw a logo? Of course, with the PGFPlot or TikZ packages already available in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, we can draw a logo. Whether a logo is good or not is certainly influenced by many factors, one of which is the creativity of the creator. Now we will discuss the steps in drawing a logo using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv class=\"bacajuga\"\u003e\n\u003ch4 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also Read\"\u003e\u003c/a\u003eAlso Read\u003c/h4\u003e\u003cul\u003e \u003cli\u003e\u003ca href=\"/2021/10/menggambar-logo-mitsubishi.html\"\u003eDrawing Mitsubishi Logo\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"/2021/10/cara-menggambar-secangkir-kopi.html\"\u003eDrawing of a Cup of Coffee\u003c/a\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"/2021/06/cara%20menggambar%20parasut%20terjun%20payung%20menggunakan%20LaTeX.html\"\u003eDrawing a Skydiving Parachute\u003c/a\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/div\u003e\n\u003ch2 id=\"typeface\" class=\"headerLink\"\u003e\n \u003ca href=\"#typeface\" class=\"header-mark\" aria-label=\"Direct link to section: Typeface\"\u003e\u003c/a\u003eTypeface\u003c/h2\u003e\u003cp\u003eThe font types used in this guide are: FrontPage-Bold.ttf, Quicksand-Bold.ttf, and TlwgTypist-Bold.otf. You can download it on the page \u003ca href=\"https://www.fontpalace.com/font-details/frontpage-bold/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://www.fontpalace.com/font-details/frontpage-bold/\u003c/a\u003e and on the page \u003ca href=\"https://linux.thai.net/pub/thailinux/software/fonts-tlwg/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://linux.thai.net/pub/thailinux/software/fonts-tlwg/\u003c/a\u003e. Place the letter file in the same folder as the document file created.\u003c/p\u003e\n\u003ch2 id=\"preamble\" class=\"headerLink\"\u003e\n \u003ca href=\"#preamble\" class=\"header-mark\" aria-label=\"Direct link to section: Preamble\"\u003e\u003c/a\u003ePreamble\u003c/h2\u003e\u003cp\u003eIn the preamble, we write down the packages needed and also the literature. As an example, you can see the following code.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\documentclass[border=3]{article}\\usepackage {tikz}\\usetikzlibrary {decorations.text,calc}\\usepackage{fontspec}\\setmainfont{FrontPage-Bold.ttf}\\newfontfamily\\quick{Quicksand-Bold.ttf}\\newfontfamily\\tlwg{TlwgTypist-Bold.otf}\u003c/code\u003e\u003c/pre\u003e\n\u003ch2 id=\"drawing-logos\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-logos\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Logos\"\u003e\u003c/a\u003eDrawing Logos\u003c/h2\u003e\u003cp\u003eTo draw a logo using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, here I use the TikZ package in the preamble. And in the body of the document, I will create a logo in the form of several circles with different colors. For details, see the following command code.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{document}\\begin{tikzpicture}\\fill[green!50](0,0) circle (3.7);\\end{tikzpicture}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThe first circle created is green, and later inside the circle I will write two texts as identities separated by a blue horizontal line. For example, the following command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\node[font=\\tlwg\\fontsize{17}{17}\\selectfont,black] at (0,0) {www.silagebra.com};\\draw[line width=3.5pt,blue](-3.6,-0.5)--(3.6,-0.5);\\node[font=\\tlwg\\fontsize{12}{12}\\selectfont,black] at (0.05,-1) {seni tipografi matematika};\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThen I creates several more circles using the following command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\draw[red!70, line width=8pt] (0,0) circle (4);\\draw[blue!60, line width=5pt] (0,0) circle (4.3);\\draw[line width=3pt] (0,0) circle (4.5);\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThe final result of the document after being compiled will produce the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEgP0IcdqTDHWsJcJgMu9VWFxe-BdO6Y1F66So0pZ0iPd5MIZQBi2Hy4ehS5vSPfuH2WfWPnoOCwsDfHE_n1VnQ2VKjfEeJesKXvbZin7uaHGhA-AliXbHLH1ZZJrfWmrl5GPBgI24SrtT4PZ1bj8LWr_wnpzIaoTQUYueYDIOzallPT9LLZe6EtJt2N=s381\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"menggambar logo menggunakan latex\" border=\"0\" data-original-height=\"312\" data-original-width=\"381\" height=\"262\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEgP0IcdqTDHWsJcJgMu9VWFxe-BdO6Y1F66So0pZ0iPd5MIZQBi2Hy4ehS5vSPfuH2WfWPnoOCwsDfHE_n1VnQ2VKjfEeJesKXvbZin7uaHGhA-AliXbHLH1ZZJrfWmrl5GPBgI24SrtT4PZ1bj8LWr_wnpzIaoTQUYueYDIOzallPT9LLZe6EtJt2N=w320-h262\" title=\"menggambar logo menggunakan latex\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat\u0026rsquo;s how to draw a logo using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Of course there are many other logos that can be drawn using LaTeX. Your creativity as a LaTeX user certainly determines the layout of the logo you draw. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Writing Learning Modules", "date_published" : "2023-03-01T07:32:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menulis-modul-pembelajaran/", "url" : "https://www.aantriono.com/en/menulis-modul-pembelajaran/", "summary": " IntroductionIn teaching activities, a teacher has many choices of teaching media. This media is used as a tool for teachers to convey lesson material well and correctly. Namely coherent, easy to understand, and also conveyed as clearly as possible in accordance with basic competencies or learning outcomes. One of the teaching media commonly used by teachers is learning modules. The definition of a module is simply defined as a form of teaching resource. Modules are prepared by a learning group or individual, which is then arranged in a neat systematic manner. Next it will be printed, this printed result will then be used as a guide for the teacher as well as for the students. The existence of this learning module then allows the entire class to continue learning. Even if the teacher who teaches in that class cannot fill in for one reason or another. So that learning activities can continue. Here I will share a guide to writing learning modules using LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn teaching activities, a teacher has many choices of teaching media. This media is used as a tool for teachers to convey lesson material well and correctly. Namely coherent, easy to understand, and also conveyed as clearly as possible in accordance with basic competencies or learning outcomes. One of the teaching media commonly used by teachers is learning modules. The definition of a module is simply defined as a form of teaching resource. Modules are prepared by a learning group or individual, which is then arranged in a neat systematic manner. Next it will be printed, this printed result will then be used as a guide for the teacher as well as for the students. The existence of this learning module then allows the entire class to continue learning. Even if the teacher who teaches in that class cannot fill in for one reason or another. So that learning activities can continue. Here I will share a guide to writing learning modules using LaTeX.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEjyVnd0QnLBG4cVRTtMRHx9bWekb5ZGK6EkvV8oi5HfSPgiUiIZsUxywCRxx4pIGVreuLMHn7VoFhm6LGd_dT03HYua4lG1So5JdAgv3o7D8GeWPTESkB4HNxe4YHgllfoy1ElRZBdUkbjZMHyABj0zlLfw5V1eRCM5oUbXQsIGEIhxTtK8ppMvZEv9=s548\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Menulis Modul Pembelajaran Matematika denga LaTeX\" border=\"0\" data-original-height=\"548\" data-original-width=\"330\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEjyVnd0QnLBG4cVRTtMRHx9bWekb5ZGK6EkvV8oi5HfSPgiUiIZsUxywCRxx4pIGVreuLMHn7VoFhm6LGd_dT03HYua4lG1So5JdAgv3o7D8GeWPTESkB4HNxe4YHgllfoy1ElRZBdUkbjZMHyABj0zlLfw5V1eRCM5oUbXQsIGEIhxTtK8ppMvZEv9=w193-h320\" title=\"Modul Pembelajaran Matematika\" width=\"193\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eAs a first step we choose a document class \u003cstrong\u003ebook\u003c/strong\u003e, and in the preamble we write the packages used along with their libraries. For example as follows.\u003c/p\u003e\n\u003ch2 id=\"preamble\" class=\"headerLink\"\u003e\n \u003ca href=\"#preamble\" class=\"header-mark\" aria-label=\"Direct link to section: Preamble\"\u003e\u003c/a\u003ePreamble\u003c/h2\u003e\u003cpre\u003e\u003ccode\u003e\\documentclass[a4paper,12pt]{book}\\usepackage[left=3cm,right=2cm,top=2cm,bottom=2cm]{geometry}\\usepackage[indonesian]{babel}\\usepackage{fourier}\\usepackage{multicol}\\usepackage{booktabs}\\usepackage{tabularx} \\usepackage[shortlabels, inline]{enumitem}\\usepackage[svgnames,dvipsnames]{xcolor}\\usepackage[many]{tcolorbox}\\tcbuselibrary{breakable} \\newtcolorbox[auto counter,number within=chapter]{thm}[2][]{%\tcolback=green!10!white,colframe=green!20!white,fonttitle=\\bfseries,coltitle=red,\ttoprule=3pt,\tleftrule=10pt,\trightrule=5pt,\ttitlerule=0pt,\ttitle=Th\\'eor\\'eme.~\\thetcbcounter} \\newtcolorbox{myfancybox}[1]{breakable,enhanced jigsaw,colback=blue!10,colframe=red,top=20pt,overlay unbroken={\\node[fill=green,text=yellow] at (frame.east) {$\\clubsuit$};},\tattach boxed title to top left={xshift=220pt,yshift=-7pt},\tboxed title style={size=small,colback=blue,colframe=red},\ttitle=#1}\\usepackage{graphicx}\\title{Modul Matematika}\\newcommand{\\booksubtitle}{Untuk kelas IX Semester 1}\\newcommand{\\booklicense}{Creative Commons Zero 1.0 Universal}\\author{Aan Triono, S.Pd}\\newcommand{\\authorsubtitle}{SMPN Purworejo, Lampung Tengah}\\makeatletter\\newcommand{\\booktitle}{\\@title}\\newcommand{\\bookauthor}{\\@author}\\makeatother \\usepackage[tikz,blur]{bclogo}\\usetikzlibrary{calc} % this is for graphics. e.g. rectangle on title page\\usepackage{amsmath,amssymb} % Used by equations\\let\\ds\\displaystyle\\renewcommand{\\contentsname}{Table of Contents} % default is {Contents}\\NewEnviron{myremark}[1]{\\par\\medskip\\noindent\t\\begin{tikzpicture}\t\t\\node[inner sep=0pt] (box) {\\parbox[t]{.99\\textwidth}{%\t\t\t\t\\begin{minipage}{.3\\textwidth}\t\t\t\t\t\\centering\\tikz[scale=5]\\node[scale=3,rotate=30]{\\bclampe};\t\t\t\t\\end{minipage}%\t\t\t\t\\begin{minipage}{.65\\textwidth}\t\t\t\t\t\\textbf{#1}\\par\\smallskip\t\t\t\t\t\\BODY\t\t\t\t\\end{minipage}\\hfill}%\t\t};\t\t\\draw[blue!75!black,line width=3pt] \t\t( $ (box.north east) + (-5pt,3pt) $ ) -- ( $ (box.north east) + (0,3pt) $ ) -- ( $ (box.south east) + (0,-3pt) $ ) -- + (-5pt,0);\t\t\\draw[blue!75!black,line width=3pt] \t\t( $ (box.north west) + (5pt,3pt) $ ) -- ( $ (box.north west) + (0,3pt) $ ) -- ( $ (box.south west) + (0,-3pt) $ ) -- + (5pt,0);\t\\end{tikzpicture}\\par\\medskip%}\\usepackage{hyperref}\\hypersetup{\tcolorlinks=true,\tlinkcolor=blue,\tfilecolor=magenta, \turlcolor=blue,}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eInformation:\u003c/p\u003e\n\u003col class=\"style1\"\u003e \u003cli\u003eThe document class used is **book, **paper **A4**, and font size **12pt**.\u003c/li\u003e\n\u003cli\u003ePackage **geometry** used to set page margins or document layout.\u003c/li\u003e\n\u003cli\u003ePackage **Babylon** used to regulate the rules of the language used.\u003c/li\u003e\n\u003cli\u003ePackage **fourier** is the type of letter used.\u003c/li\u003e\n\u003cli\u003ePackage** multicol** used to divide a page into columns.\u003c/li\u003e\n\u003cli\u003ePackage **booktabs **used to create tables.\u003c/li\u003e\n\u003cli\u003ePackage **tabularx** used to create tables.\u003c/li\u003e\n\u003cli\u003ePackage** enumitem** used to create item lists.\u003c/li\u003e\n\u003cli\u003ePackage **xcolor **used for coloring in documents.\u003c/li\u003e\n\u003cli\u003ePackage **tcolorbox** to create colored boxes in a document.\u003c/li\u003e\n\u003cli\u003ePackage **blogo** to create colored boxes in a document.\u003c/li\u003e\n\u003cli\u003ePackage **amsmath, amssymb **used to write mathematics and its symbols.\u003c/li\u003e\n\u003cli\u003ePackage **hyperref** to create a link or links in a document.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"document-body\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-body\" class=\"header-mark\" aria-label=\"Direct link to section: Document Body\"\u003e\u003c/a\u003eDocument Body\u003c/h2\u003e\u003cp\u003eEntering the body of the document we write down the parts of the learning module that are written, for example as follows.\u003c/p\u003e\n\u003cp\u003eA. Front part, which among other things consists of:\u003c/p\u003e\n\u003cul\u003e\u003cli\u003eCover\u003c/li\u003e\n\u003cli\u003eValidity sheet\u003c/li\u003e\n\u003cli\u003eForeword\u003c/li\u003e\n\u003cli\u003eList of contents\u003c/li\u003e\n\u003cli\u003eList of images (if any)\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eB. Contents section, which consists of:\u003c/p\u003e\n\u003cul\u003e\u003cli\u003eChapter or Chapters\u003c/li\u003e\n\u003cli\u003ePart or Section\u003c/li\u003e\n\u003cli\u003eSubsection or Subsection\u003c/li\u003e\n\u003cli\u003eSubsection or Subsubsection\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eC. Back\u003c/p\u003e\n\u003cul\u003e\u003cli\u003eIndex (if any)\u003c/li\u003e\n\u003cli\u003eBibliography\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eNext, start writing each chapter, section and subsection according to the material that will be used as a learning module. If you don\u0026rsquo;t want to indent the start of each paragraph, use the command \u003cem\u003e\u003cstrong\u003e\\parindent0em\u003c/strong\u003e\u003c/em\u003e in the preamble. To minimize errors that occur, frequently compile the documents you create, so that if there is an error you immediately know the cause and find a solution. As an example of a learning module written using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, Below is an example. Please click on the image below, then swipe left to see the module page by page.\u003c/p\u003e\n\u003ciframe allowfullscreen=\"true\" allowtransparency=\"true\" frameborder=\"0\" scrolling=\"no\" seamless=\"seamless\" src=\"https://online.anyflip.com/jqazz/anrh/index.html\" style=\"height: 350px; width: 550px;\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat\u0026rsquo;s the article about writing learning modules using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. If you look at the appearance, at least in my view, it looks better, neater and more elegant compared to documents typed using other tools. What do you think?\u003c/p\u003e\n", "language": "en" }, { "title" : "History of TeX and LaTeX", "date_published" : "2023-03-01T06:32:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sejarah-tex-dan-latex/", "url" : "https://www.aantriono.com/en/sejarah-tex-dan-latex/", "summary": " IntroductionTeX\\TeXTE​X is document-processing software primarily intended for producing documents that contain mathematical notation. It was created by Donald E. Knuth in May 1977 as a document formatting language.\nAt first, TeX\\TeXTE​X was implemented in Pascal, but the implementation most widely used today is generally written in C. The commands available in modern implementations have also evolved significantly compared with Knuth’s original 1977 design.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is document-processing software primarily intended for producing documents that contain mathematical notation. It was created by \u003cstrong\u003eDonald E. Knuth\u003c/strong\u003e in May 1977 as a \u003cem\u003edocument formatting language\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003eAt first, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e was implemented in Pascal, but the implementation most widely used today is generally written in C. The commands available in modern implementations have also evolved significantly compared with Knuth’s original 1977 design.\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is well known for producing high-quality documents. However, many users found its low-level commands difficult to use for structured documents containing chapters, sections, paragraphs, numbered tables, figures, and similar elements.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOakXAkB9EsngNaRsvg-518UyxWRCb37S7-pONdRmIX7LdX5JqgFdT_bZU8KM-pXAbhq8HCQPRLoRZ99nd5hCHaU6gH7989vJGwGb4lbiSi6PJvBmijrstgHx1mH1H-qMTZH33lI7a2Mo/s320/latex_logo.png' alt=\"LaTeX logo\" \u003e\u003c/p\u003e\n\u003cp\u003eTo address that problem, \u003cstrong\u003eLeslie Lamport\u003c/strong\u003e wrote a number of additional commands that run on top of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. This extension later became known as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003ePeople often pronounce \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e as “text” and “latex”. Strictly speaking, that pronunciation is not correct. The word \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e comes from the Greek letters \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eτ\u003c/mi\u003e\u003cmi\u003eϵ\u003c/mi\u003e\u003cmi\u003eχ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\tau\\epsilon\\chi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\"\u003eτ\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eϵχ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, sharing the same root as the word \u003cem\u003etechnology\u003c/em\u003e. Because of that, the intended pronunciation is closer to “tekh” and “latekh”.\u003c/p\u003e\n\u003cp\u003eWhen the stylized forms of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cannot be typeset properly, they may simply be written as \u003ccode\u003eTeX\u003c/code\u003e and \u003ccode\u003eLaTeX\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e introduced by Leslie Lamport was first known as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e 2.09. Over time, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e continued to develop. The refined version that later became widely used is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e2e, introduced in June 1994.\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e2e brought several improvements over \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e 2.09, especially in support for color, fonts, and graphics. One reason for its release was to unify various \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e formats that were not always compatible with one another, such as SLi\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, AMS\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and others.\u003c/p\u003e\n\u003ch2 id=\"advantages-and-disadvantages-of-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#advantages-and-disadvantages-of-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Advantages and Disadvantages of LaTeX\"\u003e\u003c/a\u003eAdvantages and Disadvantages of LaTeX\u003c/h2\u003e\u003cp\u003eThe standard version of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e has several advantages:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ean excellent standard for preparing text, technical formulas, and tables;\u003c/li\u003e\n\u003cli\u003eeasier use for authors once they are familiar with the workflow;\u003c/li\u003e\n\u003cli\u003edocument portability across multiple platforms;\u003c/li\u003e\n\u003cli\u003esupport for many languages;\u003c/li\u003e\n\u003cli\u003ebroad availability and free use;\u003c/li\u003e\n\u003cli\u003estrong maintenance and support from the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e3 team led by Frank Mittelbach.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eOn the other hand, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e also has several drawbacks:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eit can be difficult for beginners;\u003c/li\u003e\n\u003cli\u003eit requires care and discipline because documents are written using structured syntax;\u003c/li\u003e\n\u003cli\u003edocuments must be compiled to see the final output, so writing often involves repeated trial and revision.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat is a short overview of the history of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\TeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Understanding this background helps explain why \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e remains especially strong for scientific and mathematical documents.\u003c/p\u003e\n", "language": "en" }, { "title" : "LaTeX Learning Guide Book", "date_published" : "2022-07-19T01:31:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/buku-panduan-belajar-latex/", "url" : "https://www.aantriono.com/en/buku-panduan-belajar-latex/", "summary": " Book Description and FeaturesLaTeX is a document preparation system most suitable for producing mathematical and scientific documents with high typographic quality. LaTeX works based on the TeX program created by Donald E. Knuth. With LaTeX, writers can focus completely on the content that will be included in the document, and LaTeX handles the formatting such as tables of contents, writing formulas, table images, references and so on. Apart from being used for writing documents, LaTeX has the reliable ability to produce high-quality images such as geometric images, diagrams and graphs. We will get neat documents and charming images after using LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"book-description-and-features\" class=\"headerLink\"\u003e\n \u003ca href=\"#book-description-and-features\" class=\"header-mark\" aria-label=\"Direct link to section: Book Description and Features\"\u003e\u003c/a\u003eBook Description and Features\u003c/h2\u003e\u003cp\u003eLaTeX is a document preparation system most suitable for producing mathematical and scientific documents with high typographic quality. LaTeX works based on the TeX program created by Donald E. Knuth. With LaTeX, writers can focus completely on the content that will be included in the document, and LaTeX handles the formatting such as tables of contents, writing formulas, table images, references and so on. Apart from being used for writing documents, LaTeX has the reliable ability to produce high-quality images such as geometric images, diagrams and graphs. We will get neat documents and charming images after using LaTeX.\u003c/p\u003e\n\u003cp\u003eTherefore, this book/ebook is presented to the reader as a means of realizing this document. This ebook is also available on Google Play Book and books.google.co.id. If you want to read some of the pages, please click the preview button below or purchase directly via WhatsApp.\u003c/p\u003e\n\u003cdiv class=\"latex-book-card\"\u003e\n \u003cdiv class=\"latex-book-image-area\"\u003e\n \u003cimg alt=\"LaTeX Learning Guide Book\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirlHAq7jCb6Wk7gl8dwG_Tp7ybDtzR5sT0SDaypFfUQSQY3eLernRLulqREjQ03H_KunC3TJLq19odgvK2ZXE-OjNUQq7-YSl-a7hxHcDi3hUm_3t4AJy6siwt6kRGqDsDQNUTbD-Z3j7l4kYisrAXPNfXMZYzAj3QV63C8C_RJC-VIL5P8pF0XdaJ/s800/buku.jpeg\" class=\"latex-book-cover\" /\u003e\n \u003c/div\u003e\n \u003cdiv class=\"latex-book-details\"\u003e\n \u003cspan class=\"latex-badge\"\u003eEbook \u0026amp; Print\u003c/span\u003e\n \u003ch3 class=\"latex-card-title\"\u003eLaTeX Learning Guide Book\u003c/h3\u003e\n \u003cdiv class=\"latex-rating\"\u003e⭐⭐⭐⭐⭐ \u003cspan class=\"latex-rating-text\"\u003e(5.0/5)\u003c/span\u003e\u003c/div\u003e\n \u003cdiv class=\"latex-price\"\u003e\n \u003cspan class=\"latex-price-amount\"\u003eRp 150,000\u003c/span\u003e\n \u003c/div\u003e\n \u003cul class=\"latex-specs\"\u003e\n \u003cli\u003e\n \u003csvg class=\"latex-spec-icon\" viewBox=\"0 0 20 20\" fill=\"currentColor\"\u003e\n \u003cpath fill-rule=\"evenodd\" d=\"M10 18a8 8 0 100-16 8 8 0 000 16zm3.707-9.293a1 1 0 00-1.414-1.414L9 10.586 7.707 9.293a1 1 0 00-1.414 1.414l2 2a1 1 0 001.414 0l4-4z\" clip-rule=\"evenodd\" /\u003e\n \u003c/svg\u003e\n \u003cspan\u003eHigh typographic quality for mathematics\u003c/span\u003e\n \u003c/li\u003e\n \u003cli\u003e\n \u003csvg class=\"latex-spec-icon\" viewBox=\"0 0 20 20\" fill=\"currentColor\"\u003e\n \u003cpath fill-rule=\"evenodd\" d=\"M10 18a8 8 0 100-16 8 8 0 000 16zm3.707-9.293a1 1 0 00-1.414-1.414L9 10.586 7.707 9.293a1 1 0 00-1.414 1.414l2 2a1 1 0 001.414 0l4-4z\" clip-rule=\"evenodd\" /\u003e\n \u003c/svg\u003e\n \u003cspan\u003eProfessional geometric graphics \u0026amp; diagrams\u003c/span\u003e\n \u003c/li\u003e\n \u003cli\u003e\n \u003csvg class=\"latex-spec-icon\" viewBox=\"0 0 20 20\" fill=\"currentColor\"\u003e\n \u003cpath fill-rule=\"evenodd\" d=\"M10 18a8 8 0 100-16 8 8 0 000 16zm3.707-9.293a1 1 0 00-1.414-1.414L9 10.586 7.707 9.293a1 1 0 00-1.414 1.414l2 2a1 1 0 001.414 0l4-4z\" clip-rule=\"evenodd\" /\u003e\n \u003c/svg\u003e\n \u003cspan\u003eClean and interactive PDF formatting\u003c/span\u003e\n \u003c/li\u003e\n \u003c/ul\u003e\n \u003cdiv class=\"latex-btn-group\"\u003e\n \u003ca href=\"https://books.google.co.id/books?id=Put4EAAAQBAJ\u0026pg=PA108\u0026lpg=PA108\u0026dq=aan+triono\u0026source=bl\u0026ots=eF3s_b6_d_\u0026sig=ACfU3U2NNhRD65Bt6AQ0Re6b4_WGZp0vXA\u0026hl=en\u0026sa=X\u0026ved=2ahUKEwjDkp-lko75AhVXRmwGHR2mC004HhDoAXoECBUQAw#v=onepage\u0026q=aan%20triono\u0026f=false\" target=\"_blank\" rel=\"noopener\" class=\"latex-btn-sec\"\u003e\n \u003csvg class=\"latex-btn-svg\" viewBox=\"0 0 24 24\" fill=\"currentColor\"\u003e\n \u003cpath d=\"M19 2H5C3.9 2 3 2.9 3 4V20C3 21.1 3.9 22 5 22H19C20.1 22 21 21.1 21 20V4C21 2.9 20.1 2 19 2M19 20H5V4H7V11L9.5 8.5L12 11V4H19V20Z\"/\u003e\n \u003c/svg\u003e\n Read Preview\n \u003c/a\u003e\n \u003ca href=\"https://api.whatsapp.com/send?phone=6282376636929\u0026text=Halo%20Admin%20Saya%20mau%20pesan%20ebook%20Panduan%20Belajar%20LaTeX\" target=\"_blank\" rel=\"noopener\" class=\"latex-btn-pri\"\u003e\n \u003csvg class=\"latex-btn-svg\" viewBox=\"0 0 24 24\" fill=\"currentColor\"\u003e\n \u003cpath d=\"M.057 24l1.687-6.163c-1.041-1.804-1.588-3.849-1.587-5.946C.06 5.348 5.397.01 12.008.01c3.202.001 6.212 1.246 8.477 3.514 2.266 2.268 3.507 5.28 3.505 8.484-.004 6.657-5.34 11.997-11.953 11.997-2.005-.001-3.973-.502-5.724-1.455L0 24zm6.59-4.846c1.6.95 3.188 1.449 4.625 1.451 5.437.002 9.861-4.416 9.864-9.852.002-2.634-1.02-5.11-2.883-6.976C16.392 1.96 13.917 1.938 12.015 1.938c-5.441 0-9.866 4.418-9.869 9.856-.001 1.77.476 3.499 1.38 5.03l-.403 1.472.451-.418zm12.516-7.855c-.29-.146-1.72-.85-1.985-.946-.266-.097-.46-.144-.654.146-.193.291-.747.946-.917 1.14-.17.194-.34.218-.63.073-.29-.147-1.227-.452-2.337-1.443-.864-.771-1.447-1.723-1.616-2.016-.17-.29-.018-.447.127-.592.13-.13.29-.34.436-.51.145-.17.194-.291.291-.485.097-.194.048-.364-.024-.51-.073-.146-.654-1.576-.896-2.158-.236-.569-.475-.49-.654-.5l-.56-.007c-.193 0-.509.073-.776.364-.266.291-1.02.996-1.02 2.43 0 1.434 1.043 2.82 1.188 3.015.145.194 2.055 3.14 4.978 4.407.695.302 1.237.482 1.661.617.698.222 1.334.191 1.838.116.56-.083 1.72-.704 1.962-1.385.243-.68.243-1.264.17-1.386-.073-.121-.267-.194-.559-.34z\"/\u003e\n \u003c/svg\u003e\n Order via WhatsApp\n \u003c/a\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n", "language": "en" }, { "title" : "Similarity and Congruence Quiz", "date_published" : "2022-03-13T03:27:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kuis-materi-kesebangunan-dan/", "url" : "https://www.aantriono.com/en/kuis-materi-kesebangunan-dan/", "summary": " IntroductionOn this occasion, I made a post about a quiz on congruence and congruence material in the form of multiple choice questions. Previously, students had studied the material congruence and congruence\n. This material is the first material taught in class 9 of SMP/MTS this even semester.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eOn this occasion, I made a post about a quiz on congruence and congruence material in the form of multiple choice questions. Previously, students had studied the material \u003ca href=\"https://www.aantriono.com/2020/11/kesebangunan-dan-kekongruenan.html\" target=\"_blank\"\u003econgruence and congruence\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e. This material is the first material taught in class 9 of SMP/MTS this even semester.\u003c/p\u003e\n\u003ch2 id=\"quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Quiz\"\u003e\u003c/a\u003eQuiz\u003c/h2\u003e\u003cp\u003eThis quiz is given as a learning medium during a time when face-to-face learning is limited due to the Covid-19 pandemic, as well as a means of training students\u0026rsquo; skills in mastering IT in the 21st century. Students take the following quiz directly on this blog with the note that what is presented here is specifically for 10-item multiple choice questions.\u003c/p\u003e\n\u003cdiv class=\"postRelated\"\u003e\n \u003c!--[ Related title ]--\u003e\n\u003ch2 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also read\"\u003e\u003c/a\u003eAlso read\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-interaktif-penilaian-tengah.html\" target=\"_blank\"\u003eOdd Midterm Assessment\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-bilangan-berpangkat.html\" target=\"_blank\"\u003eRanked Numbers Material Quiz\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-persamaan-kuadrat.html\" target=\"_blank\"\u003ePower Equation Material Quiz\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n \u003c/ul\u003e\n\u003c/div\u003e\n\u003ch3 id=\"instructions-for-taking-the-quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#instructions-for-taking-the-quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Instructions for Taking the Quiz\"\u003e\u003c/a\u003eInstructions for Taking the Quiz\u003c/h3\u003e\u003col class=\"style1\"\u003e\n \u003cli\u003eRead the prayer first before taking the quiz.\u003c/li\u003e\n \u003cli\u003eClick the button Start to start the quiz.\u003c/li\u003e\n \u003cli\u003eWrite Name And Class you in the box provided.\u003c/li\u003e\n \u003cli\u003eClick the button Do it to go to the question menu.\u003c/li\u003e\n \u003cli\u003eClick on the button which is the correct answer.\u003c/li\u003e\n \u003cli\u003eClick on the sign \u003e to go to the next question, and click the sign \u003c to return to the previous question, or click question menu to see the whole question.\u003c/li\u003e\n \u003cli\u003eThe quiz time is: 30 minutes.\u003c/li\u003e\n \u003cli\u003eThe KKM for this quiz is 70\u003c/li\u003e\n \u003cli\u003eClick the button Finished to end the quiz.\u003c/li\u003e\n \u003cli\u003eMark You will appear after clicking the finish button.\u003c/li\u003e\n \u003c/ol\u003e\n\u003c!--Tombol Mulai--\u003e\n\u003cdiv class=\"center\"\u003e\n \u003cbutton id=\"klik-mulai\"\u003eStart\u003c/button\u003e\n\u003c!--Box Input Nama dan Kelas--\u003e\n\u003cdiv id=\"input-data\"\u003e\n\u003c!--Box Soal--\u003e\n\u003cdiv id=\"bank-soal\"\u003e\n\u003cdiv class=\"waktu-wrap\"\u003e\n\u003cdiv class=\"data-peserta\"\u003e\n\u003cdiv class=\"nama-peserta\"\u003e\n\u003cdiv class=\"kelas-peserta\"\u003e\n\u003c!--Waktu Soal, Ganti berdasarkan jam--\u003e\n\u003cdiv data-waktu=\"0.5\" id=\"waktu-soal\"\u003e\n \u003c!--Jumlah Soal, Ganti berdasrkan jumlah soal--\u003e\n\u003cdiv class=\"bank-soal-wrap\" data-soal=\"10\"\u003e\n\u003c!--Pertanyaan dan Jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e1. The shapes below must be similar, *except*...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003etwo squares\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003etwo rectangles\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003etwo circles\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003etwo equilateral triangles\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Penutup kode pertanyaan dan jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e2. The size of a rectangle similar to a rectangle of size $12\\ \\text{cm}\\times 4\\ \\text{cm}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$4\\ \\text{cm}\\times 2\\ \\text{cm}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$8\\ \\text{cm}\\times 3\\ \\text{cm}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$18\\ \\text{cm}\\times 6\\ \\text{cm}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$20\\ \\text{cm}\\times 8\\ \\text{cm}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e3. Look at the picture below. If $\\triangle ABC$ and $\\triangle DEF$ are congruent, the correct statement is...\u003c/p\u003e\n \u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8zqe6l_uaYyoAOjl2JrdBwIIDrKQRkQ1oAC6JBrk0iMtoOQbBrYczmuMKDmPMXSL5k4nqiGpuzruEAbbslE8KYfHh_GAY31wRdDFegGUjaB09pR0SOdpqE_WBx3qr-XSABQbd5fBCU0iajBaQKIP_9K0H_0oNClCMqOu9rtSoxPZZo-rctevD-dDU/s1280/kuis.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Segitiga yang sebangun dan kongruen\" border=\"0\" data-original-height=\"400\" data-original-width=\"1280\" height=\"125\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8zqe6l_uaYyoAOjl2JrdBwIIDrKQRkQ1oAC6JBrk0iMtoOQbBrYczmuMKDmPMXSL5k4nqiGpuzruEAbbslE8KYfHh_GAY31wRdDFegGUjaB09pR0SOdpqE_WBx3qr-XSABQbd5fBCU0iajBaQKIP_9K0H_0oNClCMqOu9rtSoxPZZo-rctevD-dDU/w400-h125/kuis.png\" title=\"Segitiga yang sebangun dan kongruen\" width=\"400\" /\u003e\u003c/a\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$AC=DF$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$AB:DE=BC:EF$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$AB\\times AC=FD\\times ED$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$AC:AB=DE:DF$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e4. A tree casts a shadow $15$ meters long on a horizontal ground surface. Meanwhile, a pole that is $5.2\\ \\text{meter}$ has a shadow of $6.5\\ \\text{meter}$. The actual height of the tree is ... meters.\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$10$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$10.5$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$11.2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$12$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e5. Look at the picture below. What is the value of $x$?\u003c/p\u003e\n \u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEi1FTP2ZY515atLKW2CZOvhJePSquGRlBzY_8CBmF9MfPEzoP2JSbcME9xjDfVCXG4TDUTblViYblwRMODjtEOl6D-CLKXzd4oe-MInRmbJ166AQSt889a_DD0QKpahHy22WHUasDXFRek2eJjT-tSw2Iv9ume4sYFI9_wbU07IlMziJiRid74-ZbT_=s229\" style=\"display: block; padding: 1em 0px; text-align: center;\"\u003e\u003cimg alt=\"\" border=\"0\" data-original-height=\"224\" data-original-width=\"229\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEi1FTP2ZY515atLKW2CZOvhJePSquGRlBzY_8CBmF9MfPEzoP2JSbcME9xjDfVCXG4TDUTblViYblwRMODjtEOl6D-CLKXzd4oe-MInRmbJ166AQSt889a_DD0QKpahHy22WHUasDXFRek2eJjT-tSw2Iv9ume4sYFI9_wbU07IlMziJiRid74-ZbT_=s320\" width=\"200\" /\u003e\u003c/a\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$6.7$ cm\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$5.0$ cm\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$4.1$ cm\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3.8$ cm\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e6. In the image below, it is known that $AB=6\\ \\text{cm}$ and $BC=8\\ \\text{cm}$. What is the length of $AD$? \u003c/p\u003e\n \u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEhNQEfq_4Z74jgRBUs3oLdcUcsvJjPwh19jaPYI-sHkhbXOMXDgnIsGYNe5g6hjaIzHruNlgggg5afwUGpfYryhNWzFB3as-3kradcINmrOZt3QXv36F89NTh6jF0zjTpkQNXXfdel0wMK631aJerGFDd3mcCzEyanx8SScTDtStUTFr0GaMgprcklf=s284\" style=\"display: block; padding: 1em 0px; text-align: center;\"\u003e\u003cimg alt=\"\" border=\"0\" data-original-height=\"284\" data-original-width=\"263\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEhNQEfq_4Z74jgRBUs3oLdcUcsvJjPwh19jaPYI-sHkhbXOMXDgnIsGYNe5g6hjaIzHruNlgggg5afwUGpfYryhNWzFB3as-3kradcINmrOZt3QXv36F89NTh6jF0zjTpkQNXXfdel0wMK631aJerGFDd3mcCzEyanx8SScTDtStUTFr0GaMgprcklf=s320\" /\u003e\u003c/a\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3.8$ cm\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3.7$ cm\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$3.6$ cm\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3.5$ cm\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e7. In the image below, the lengths $AB=CD=8\\ \\text{cm}$, and $AD=8\\ \\text{cm}$ are known. The length of $BC$ is ... cm.\u003c/p\u003e\n \u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEhbrDtOw7vOpMZ9o9_IFpIVZuGcJUjWBlYbxXauXSCCxC9boi-8i3FQ9-MNobBxhvmHnfo3HlJZj9DZhk891swYZ0Pazfz1W2a3pGRzg1jkbbIUUinZ-gWtfgSsco3jCYdzUHQOMhiTfvnBqDmPgzoXpFtHyl4dVyU7nujtq34qv4d_cJ8uaq3R8SuL=s292\" style=\"display: block; padding: 1em 0px; text-align: center;\"\u003e\u003cimg alt=\"\" border=\"0\" data-original-height=\"284\" data-original-width=\"292\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEhbrDtOw7vOpMZ9o9_IFpIVZuGcJUjWBlYbxXauXSCCxC9boi-8i3FQ9-MNobBxhvmHnfo3HlJZj9DZhk891swYZ0Pazfz1W2a3pGRzg1jkbbIUUinZ-gWtfgSsco3jCYdzUHQOMhiTfvnBqDmPgzoXpFtHyl4dVyU7nujtq34qv4d_cJ8uaq3R8SuL=s200\" width=\"200\" /\u003e\u003c/a\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$12$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$13$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$14$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$15$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e8. A photo ~~**Mamas Tamvan**~~ measuring $12\\ \\text{cm}\\times 15\\ \\text{cm}$ placed on a cardboard. At the top, left and right of the photo there is still cardboard remaining $2\\ \\text{cm}$ wide. If the photo and cardboard are similar, what is the area of ​​the cardboard? \u003c/p\u003e\n \u003ca href=\"https://blogger.googleusercontent.com/img/a/AVvXsEjbCaLXizEQAKDYIY-ohV7EXp3PdOkIpTyd7PBptDUse2nP-jUGzoxQC5V5gQZvZGWJ9fbkkFpMoIPHSYEp3gVwAXfptMiIEOwEPfNldrbkiz_iX9PIe9HymyZG-INiIpm6o0jlKEp8vmR_i_xF6WEKchArvmcRMfiDAgj7VU10jGHs1GXg6FR-F0Oc=s348\" style=\"display: block; padding: 1em 0px; text-align: center;\"\u003e\u003cimg alt=\"\" border=\"0\" data-original-height=\"348\" data-original-width=\"272\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/a/AVvXsEjbCaLXizEQAKDYIY-ohV7EXp3PdOkIpTyd7PBptDUse2nP-jUGzoxQC5V5gQZvZGWJ9fbkkFpMoIPHSYEp3gVwAXfptMiIEOwEPfNldrbkiz_iX9PIe9HymyZG-INiIpm6o0jlKEp8vmR_i_xF6WEKchArvmcRMfiDAgj7VU10jGHs1GXg6FR-F0Oc=s200\" /\u003e\u003c/a\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$300\\ \\text{cm}^2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$310\\ \\text{cm}^2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$320\\ \\text{cm}^2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$330\\ \\text{cm}^2$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e9. A model airplane is $40$ cm long and $32$ cm wide. If the actual length is $30$ m, how many meters is the actual width of the plane?\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$22$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$24$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$28$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$30$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e10. Two right triangles will be congruent if in both ...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003eThe sides of a right angle are the same length\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003ehypotenuse sides are the same length\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003ethe other two angles are equal\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003eA, B, and C are correct\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Tombol Soal, Sesuaikan dengan jumlah soal--\u003e\n\u003cdiv id=\"show-hide-box-soal\"\u003e\n\u003cdiv class=\"tombol-show-hide lihat-soal\"\u003e\n closed\n\u003cdiv class=\"menu-tombol-soal\"\u003e\n\u003cdiv class=\"tombol-soal\" data-nomor=\"1\"\u003e1\n\u003cdiv class=\"tombol-soal\" data-nomor=\"2\"\u003e2\n\u003cdiv class=\"tombol-soal\" data-nomor=\"3\"\u003e3\n\u003cdiv class=\"tombol-soal\" data-nomor=\"4\"\u003e4\n\u003cdiv class=\"tombol-soal\" data-nomor=\"5\"\u003e5\n\u003cdiv class=\"tombol-soal\" data-nomor=\"6\"\u003e6\n\u003cdiv class=\"tombol-soal\" data-nomor=\"7\"\u003e7\n\u003cdiv class=\"tombol-soal\" data-nomor=\"8\"\u003e8\n\u003cdiv class=\"tombol-soal\" data-nomor=\"9\"\u003e9\n\u003cdiv class=\"tombol-soal\" data-nomor=\"10\"\u003e10\n\u003cdiv id=\"menu-soal\"\u003e\n\u003cdiv class=\"next-prev-soal\"\u003e\n\u003cdiv class=\"prev\" id=\"prev-soal\"\u003e\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M15.41,16.58L10.83,12L15.41,7.41L14,6L8,12L14,18L15.41,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n Previously\n\u003cdiv class=\"next\" id=\"next-soal\"\u003e\n Furthermore\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M8.59,16.58L13.17,12L8.59,7.41L10,6L16,12L10,18L8.59,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n\u003cdiv class=\"menu-pilihan-soal\"\u003e\n \u003cbutton class=\"lihat-soal\" id=\"lihat-soal\"\u003eQuestion Menu\u003c/button\u003e\n\u003cdiv class=\"tombol-soal-selesai\"\u003e\n \u003cbutton id=\"soal-selesai\"\u003eFinished\u003c/button\u003e\n\u003cdiv id=\"lihat-hasil-nilai\"\u003e\n\u003cdiv id=\"hasil-pengerjaan\"\u003e\n\u003cp class=\"info-keterangan-hasil\"\u003e*Screenshot and send your score via WhatsApp or Telegram \u003ca href=\"https://www.aantriono.com/p/contact.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\n\u003cp\u003e, if remedial, you can repeat it again by clicking \u003ca href=\"https://www.aantriono.com/2022/03/kuis-materi-kesebangunan-dan.html\"\u003erestart\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e.\u003c/p\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat\u0026rsquo;s the article about quiz material on congruency and congruence for class 9 SMP/MTs in the form of multiple choice questions that I can share. If there are errors, please correct them, and if you have input, please leave them in the comments. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Data Collection and Presentation", "date_published" : "2022-02-14T16:02:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/penyajian-dan-pengumpulan-data/", "url" : "https://www.aantriono.com/en/penyajian-dan-pengumpulan-data/", "summary": " IntroductionData is essential in everyday life. Clear data helps us make decisions, compare information, and explain a situation more accurately. To make data easier to read, it is usually presented in the form of tables or diagrams.\nCollecting DataIn general, data can be divided into two main types:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eData is essential in everyday life. Clear data helps us make decisions, compare information, and explain a situation more accurately. To make data easier to read, it is usually presented in the form of tables or diagrams.\u003c/p\u003e\n\u003ch2 id=\"collecting-data\" class=\"headerLink\"\u003e\n \u003ca href=\"#collecting-data\" class=\"header-mark\" aria-label=\"Direct link to section: Collecting Data\"\u003e\u003c/a\u003eCollecting Data\u003c/h2\u003e\u003cp\u003eIn general, data can be divided into two main types:\u003c/p\u003e\n\u003ch3 id=\"quantitative-data\" class=\"headerLink\"\u003e\n \u003ca href=\"#quantitative-data\" class=\"header-mark\" aria-label=\"Direct link to section: Quantitative data\"\u003e\u003c/a\u003eQuantitative data\u003c/h3\u003e\u003cp\u003eQuantitative data is numerical data, for example:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eaverage monthly income\u003c/li\u003e\n\u003cli\u003epopulation size\u003c/li\u003e\n\u003cli\u003enumber of children in a family\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eQuantitative data can be divided into:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ediscrete data\u003c/strong\u003e, obtained by counting\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003econtinuous data\u003c/strong\u003e, obtained by measurement\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExamples of discrete data: number of students in a school.\u003cbr\u003e\nExamples of continuous data: height and weight.\u003c/p\u003e\n\u003ch3 id=\"qualitative-data\" class=\"headerLink\"\u003e\n \u003ca href=\"#qualitative-data\" class=\"header-mark\" aria-label=\"Direct link to section: Qualitative data\"\u003e\u003c/a\u003eQualitative data\u003c/h3\u003e\u003cp\u003eQualitative data is non-numerical data, for example:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eeconomic level\u003c/li\u003e\n\u003cli\u003eoccupation\u003c/li\u003e\n\u003cli\u003egender\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eAn important note:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eRange\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmtext\u003elargest value\u003c/mtext\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmtext\u003esmallest value\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\text{Range} = \\text{largest value} - \\text{smallest value}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eRange\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003elargest value\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003esmallest value\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"presenting-data\" class=\"headerLink\"\u003e\n \u003ca href=\"#presenting-data\" class=\"header-mark\" aria-label=\"Direct link to section: Presenting Data\"\u003e\u003c/a\u003ePresenting Data\u003c/h2\u003e\u003cp\u003eCollecting data is not enough by itself. Data must also be presented in a form that is easy to read and understand. Common forms include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003etables\u003c/li\u003e\n\u003cli\u003ebar charts\u003c/li\u003e\n\u003cli\u003eline charts\u003c/li\u003e\n\u003cli\u003epie charts\u003c/li\u003e\n\u003cli\u003epictograms\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"presenting-data-using-tables\" class=\"headerLink\"\u003e\n \u003ca href=\"#presenting-data-using-tables\" class=\"header-mark\" aria-label=\"Direct link to section: Presenting Data Using Tables\"\u003e\u003c/a\u003ePresenting Data Using Tables\u003c/h2\u003e\u003ch3 id=\"single-frequency-table\" class=\"headerLink\"\u003e\n \u003ca href=\"#single-frequency-table\" class=\"header-mark\" aria-label=\"Direct link to section: Single Frequency Table\"\u003e\u003c/a\u003eSingle Frequency Table\u003c/h3\u003e\u003cp\u003eConsider the following data showing the number of children in \u003ccode\u003e40\u003c/code\u003e families:\u003c/p\u003e\n\u003cp\u003e\u003ccode\u003e1, 4, 3, 4, 5, 4, 3, 6, 1, 2, 2, 3, 2, 4, 1, 6, 5, 3, 4, 3, 4, 4, 5, 4, 4, 4, 6, 5, 4, 4, 2, 4, 3, 3, 2, 4, 2, 3, 4, 1\u003c/code\u003e\u003c/p\u003e\n\u003cp\u003eA single frequency table for this data is:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eNumber of children\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eTally\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eFrequency\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e6\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eTotal\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003cstrong\u003e40\u003c/strong\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eIn a single frequency table, each row represents only one data value.\u003c/p\u003e\n\u003ch3 id=\"grouped-frequency-table\" class=\"headerLink\"\u003e\n \u003ca href=\"#grouped-frequency-table\" class=\"header-mark\" aria-label=\"Direct link to section: Grouped Frequency Table\"\u003e\u003c/a\u003eGrouped Frequency Table\u003c/h3\u003e\u003cp\u003eNow consider the following mathematics test scores:\u003c/p\u003e\n\u003cp\u003e\u003ccode\u003e44, 54, 85, 92, 73, 99, 91, 96, 74, 75, 70, 57, 83, 49, 57, 52, 64, 73, 82, 90, 70, 89, 91, 67, 52, 64, 73, 82, 59, 65, 79, 82, 89, 53, 52, 50\u003c/code\u003e\u003c/p\u003e\n\u003cp\u003eThe highest score is \u003ccode\u003e99\u003c/code\u003e and the lowest is \u003ccode\u003e44\u003c/code\u003e, so the range is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e99\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e44\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e55\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n99 - 44 = 55\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e99\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e44\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e55\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eBecause the range is fairly large, the data is more practical in a grouped frequency table:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eScore interval\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eTally\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eFrequency\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e44 - 51\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e52 - 59\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e60 - 67\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e68 - 75\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e76 - 83\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e84 - 91\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e92 - 99\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e`\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eTotal\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003cstrong\u003e36\u003c/strong\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eImportant terms:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eclass interval\u003c/strong\u003e: one score group\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003enumber of class intervals\u003c/strong\u003e: total number of groups\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003einterval length\u003c/strong\u003e: width of each interval\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eFor the table above:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003enumber of class intervals = \u003ccode\u003e7\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003einterval length = \u003ccode\u003e8\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eNotes:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eThe smallest and largest values must still be included in the intervals.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThe number of classes can be estimated using Sturges\u0026rsquo; rule:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3.3\u003c/mn\u003e\u003cmi\u003elog\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n k = 1 + 3.3 \\log n\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003elo\u003cspan style=\"margin-right:0.01389em;\"\u003eg\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the number of data values.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"presenting-data-using-diagrams\" class=\"headerLink\"\u003e\n \u003ca href=\"#presenting-data-using-diagrams\" class=\"header-mark\" aria-label=\"Direct link to section: Presenting Data Using Diagrams\"\u003e\u003c/a\u003ePresenting Data Using Diagrams\u003c/h2\u003e\u003cp\u003eData can also be displayed with diagrams.\u003c/p\u003e\n\u003ch3 id=\"bar-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#bar-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Bar Chart\"\u003e\u003c/a\u003eBar Chart\u003c/h3\u003e\u003cp\u003eA bar chart uses:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ea horizontal axis for categories\u003c/li\u003e\n\u003cli\u003ea vertical axis for values or frequencies\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExample of average mathematics scores:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eStudent\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eAverage score\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eAbdi\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e65\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eAndi\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e85\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eBayu\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e70\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eCindi\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e80\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eClara\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e60\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eDiva\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e90\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eSteps to make a bar chart:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ewrite the title\u003c/li\u003e\n\u003cli\u003eplace the categories on the horizontal axis\u003c/li\u003e\n\u003cli\u003eplace the values or frequencies on the vertical axis\u003c/li\u003e\n\u003cli\u003echoose a suitable scale\u003c/li\u003e\n\u003cli\u003edraw one bar for each category\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLrYvLoy9gWDMjlWXVtOtQuKDzoEp366bPhbytjKV4Nd8rkFaTe44W1VGok8JZdK5vy9CdNQfupHHq99alRsdhXon91FxLxsH5EcdvNvF1spD41sicw-smGunSaHe4iemkEfLJhACHMJVrGMCq5SFnROBJ1fOK-EB8zCfuKO0bw284khNVRFhkcesF/s1280/diagram%20(1).png' alt=\"Bar chart\" \u003e\u003c/p\u003e\n\u003cp\u003eIn a bar chart, all bars have the same width, while the height represents the value or frequency.\u003c/p\u003e\n\u003ch3 id=\"pie-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#pie-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Pie Chart\"\u003e\u003c/a\u003ePie Chart\u003c/h3\u003e\u003cp\u003eA pie chart presents data as sectors of a circle. The central angle of each sector represents the proportion of one category compared with the entire data set.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cp\u003eIn one class of \u003ccode\u003e60\u003c/code\u003e students, each student chooses one extracurricular activity:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ebasketball: \u003ccode\u003e15\u003c/code\u003e students\u003c/li\u003e\n\u003cli\u003evolleyball: \u003ccode\u003e17\u003c/code\u003e students\u003c/li\u003e\n\u003cli\u003ePMR: \u003ccode\u003e24\u003c/code\u003e students\u003c/li\u003e\n\u003cli\u003escouts: \u003ccode\u003e4\u003c/code\u003e students\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe calculations are:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eActivity\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eFrequency\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003ePercentage\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eCentral angle\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eBasketball\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e25%\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e90°\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eVolleyball\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e17\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e28.3%\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e102°\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003ePMR\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e24\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e40%\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e144°\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eScouts\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e6.7%\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e24°\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cstrong\u003eTotal\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003cstrong\u003e60\u003c/strong\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e\u003cstrong\u003e360°\u003c/strong\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eFor scouts:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{4}{60} \\times 360^\\circ = 24^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/a/AVvXsEgVILL7wlqYbid33RxDj8Y_QiFzwS-8qPM-Lm7Y9UmiKFJAFR92NaZAkGFhPp4evWFoUFDGT22iCWCB_gT78bCKokabrOsYn8RRJ0g19y6cf08YM15daFHoV3YRZOWJg102HUt9MWhyXbDLyUGBi6KAizEv2_X7axI6XOzw1A4Z_jIaspoWMNSc6TJm=s417' alt=\"Pie chart\" \u003e\u003c/p\u003e\n\u003ch3 id=\"line-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#line-chart\" class=\"header-mark\" aria-label=\"Direct link to section: Line Chart\"\u003e\u003c/a\u003eLine Chart\u003c/h3\u003e\u003cp\u003eA line chart is commonly used to show change over time, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003edaily visitors\u003c/li\u003e\n\u003cli\u003ehourly temperature\u003c/li\u003e\n\u003cli\u003emonthly sales\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eData points are plotted on a coordinate plane and then connected with line segments.\u003c/p\u003e\n\u003ch3 id=\"pictogram\" class=\"headerLink\"\u003e\n \u003ca href=\"#pictogram\" class=\"header-mark\" aria-label=\"Direct link to section: Pictogram\"\u003e\u003c/a\u003ePictogram\u003c/h3\u003e\u003cp\u003eA pictogram presents data using pictures or symbols. Each picture stands for a certain number of objects.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e1 book icon may represent 5 books\u003c/li\u003e\n\u003cli\u003e1 student icon may represent 10 students\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003ePictograms are useful for simple data that should be understood quickly.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eCollecting and presenting data is a basic part of statistics. When data is presented in the right format, it becomes much easier to read, compare, and analyze.\u003c/p\u003e\n", "language": "en" }, { "title" : "How to Embed GeoGebra into a Blog", "date_published" : "2021-10-26T13:11:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-embed-geogebra-ke-dalam-blog/", "url" : "https://www.aantriono.com/en/cara-embed-geogebra-ke-dalam-blog/", "summary": " IntroductionGeoGebra is dynamic software that is very useful for teaching mathematics, especially geometry and algebra. It continues to evolve, and it is now available as GeoGebra 6. Geometric figures created in GeoGebra can be exported to LaTeX\\LaTeXLATE​X, while GeoGebra applets can be embedded into Moodle, Google Classroom, blogs, or websites.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eGeoGebra is dynamic software that is very useful for teaching mathematics, especially geometry and algebra. It continues to evolve, and it is now available as GeoGebra 6. Geometric figures created in GeoGebra can be exported to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, while GeoGebra applets can be embedded into Moodle, Google Classroom, blogs, or websites.\u003c/p\u003e\n\u003cp\u003eIn this article, I will show how to embed GeoGebra into a blog. Before starting, make sure you already have an account at \u003ccode\u003ehttps://www.geogebra.org\u003c/code\u003e.\u003c/p\u003e\n\u003ch4 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also Read\"\u003e\u003c/a\u003eAlso Read\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003e\u003ca href=\"https://www.silagebra.com/2021/03/animasi-jam.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eClock Animation\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.silagebra.com/2021/05/menggambar-segitiga.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eDrawing Triangles\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2020/12/unsur-unsur-lingkaran.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eCircle Elements\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"embed-guide\" class=\"headerLink\"\u003e\n \u003ca href=\"#embed-guide\" class=\"header-mark\" aria-label=\"Direct link to section: Embed Guide\"\u003e\u003c/a\u003eEmbed Guide\u003c/h2\u003e\u003cp\u003eThe steps for embedding GeoGebra into a blog are as follows.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eOpen \u003ccode\u003ehttps://www.geogebra.org\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eClick \u003ccode\u003eSign in\u003c/code\u003e in the top-right corner.\u003c/li\u003e\n\u003cli\u003eAfter signing in, click \u003ccode\u003eProfile\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/a/AVvXsEg2Mi96RUqa-kC2nkKGhE-jy11Ws24JVlFUWd9F-ykXy20YqoCicAD3e_FAkP6Ou6NSLyZ1Pv6x1IyRjpSdfe_K8zlR7VdLjuFUCY4xWDPNHkaaq3LKdEiVBhiZ2ZeQ-42o7nj462S6KorZVzof-WH6PxMl7F8sJ42LP88TpMM0Ksii6cvJWqB2QiHv=s760' alt=\"GeoGebra profile page\" \u003e\u003c/p\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003e\n\u003cp\u003eSelect the \u003ccode\u003eActivity\u003c/code\u003e applet that you want to embed into the blog. The applet can be created first or uploaded from a file you made earlier.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eClick the three-dot menu to the right of the selected \u003ccode\u003eActivity\u003c/code\u003e, then click \u003ccode\u003eDetails\u003c/code\u003e.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/a/AVvXsEinfiXEU9hhgB5I_F3oZ2yjUqIQYeadu3gOCZ8tEaB2o2lWVq2KXKcQJ3LxKotTQOFol-Yn8LrcYSPBamWBZ07GETVGd2LTUpsTljvbr6-3e8jfU_bCR8s4KlOytM_0R6ELYgCXglUSVdZgemUSR_hChyIdNXKpC8bUsEAmL8Q90kUxw98UC-uOr2ut=s261' alt=\"GeoGebra activity details menu\" \u003e\u003c/p\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eClick \u003ccode\u003eShare\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/a/AVvXsEhfh3WT6rIJeSXXKeWyGWgIVg11zBG_tahI87n5n8dzUcY6xuow97Oz83p395IXX2iofeZ3IvAemvf3XVcy3ruQpEQiVoPwhpuNCirsA_BAqF9rJC6-c4XdcHzBDRATkAkcOQIYBnmLSJO3kDe9suC-r7ePvRlFh-K3yMKeIdd3Xy2bDwu47kAirTK7=s1273' alt=\"GeoGebra share button\" \u003e\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eClick \u003ccode\u003eEmbed\u003c/code\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/a/AVvXsEjzd2kvhb0SFtRFWgZzY7C3V9H_spuXbjtvZJ4YOATjK2xUJ1BlmHmJGewc8UrsuV9ZGEA-PqgnmJ1UVqawO1lgQDpThXG8mxzNxybEtHZakv74knY-wz5YD3keQQVKLPNsLZ__0tYhvIKmeejc49gkrPbMP80nVohe16VYousf1C-qLHvXbSDp6IwA=s713' alt=\"GeoGebra embed menu\" \u003e\u003c/p\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eCopy the HTML code shown in the box.\u003c/li\u003e\n\u003cli\u003eLog in to \u003ccode\u003eblogger.com\u003c/code\u003e or your blog dashboard.\u003c/li\u003e\n\u003cli\u003eSelect the post where you want to insert GeoGebra.\u003c/li\u003e\n\u003cli\u003eSwitch the editor to \u003ccode\u003eHTML\u003c/code\u003e mode.\u003c/li\u003e\n\u003cli\u003ePaste the code in the desired location. An example of the embedded result is shown below.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/mjdvyccs/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Keliling dan Luas Lingkaran\" width=\"650px\"\u003e \u003c/iframe\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThose are the steps for embedding GeoGebra into a blog. If there are any errors, please leave corrections, and if you have suggestions, please write them in the comments. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Algebraic Forms and Operations", "date_published" : "2021-10-20T07:48:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bentuk-aljabar-dan-operasinya/", "url" : "https://www.aantriono.com/en/bentuk-aljabar-dan-operasinya/", "summary": " Introduction\nAl-Khwarizmi is widely known as one of the major figures in the history of algebra. In school mathematics, algebraic forms are used to represent unknown quantities with letters or variables.\nSimple examples:\nthe square of 333 is 32=93^2 = 932=9 the square root of 999 is 9=3\\sqrt{9} = 39​=3 A Problem Involving Algebraic FormsA traveler rides for 3 hours at an average speed of (2x−5)(2x-5)(2x−5) km/h, then continues for 4 hours at an average speed of (4x+8)(4x+8)(4x+8) km/h. If the total distance traveled is 346 km, the value of xxx can be found using algebraic expressions.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/a/AVvXsEjp6WxPsQPkpq63-MalsGS3eno3fwHskKiBbzEwjLfc8urWYUB0FLzK9t6_UgBZ_tC7dDj0ayqIBr65czq6gqURwS8Q2pfUb0jWKFiZmrvIy0GbIIckPoRx5N1f57DcXn9dQQ9_bRKxKjs5kYsTAmCpCwIXHrvq7lCgHp6JiiJZl-2iHVTe6vBDDQK2=s1280' alt=\"Algebraic forms and operations\" \u003e\u003c/p\u003e\n\u003cp\u003eAl-Khwarizmi is widely known as one of the major figures in the history of algebra. In school mathematics, algebraic forms are used to represent unknown quantities with letters or variables.\u003c/p\u003e\n\u003cp\u003eSimple examples:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe square of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^2 = 9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003ethe square root of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{9} = 3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"a-problem-involving-algebraic-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-problem-involving-algebraic-forms\" class=\"header-mark\" aria-label=\"Direct link to section: A Problem Involving Algebraic Forms\"\u003e\u003c/a\u003eA Problem Involving Algebraic Forms\u003c/h2\u003e\u003cp\u003eA traveler rides for \u003ccode\u003e3\u003c/code\u003e hours at an average speed of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2x-5)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e km/h, then continues for \u003ccode\u003e4\u003c/code\u003e hours at an average speed of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(4x+8)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e km/h. If the total distance traveled is \u003ccode\u003e346\u003c/code\u003e km, the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e can be found using algebraic expressions.\u003c/p\u003e\n\u003ch2 id=\"definition-of-an-algebraic-form\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-of-an-algebraic-form\" class=\"header-mark\" aria-label=\"Direct link to section: Definition of an Algebraic Form\"\u003e\u003c/a\u003eDefinition of an Algebraic Form\u003c/h2\u003e\u003cp\u003eAn algebraic form is a mathematical expression that uses letters to represent unknown numbers.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5a + 6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x - 1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2 + 2x + 3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"elements-of-algebraic-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#elements-of-algebraic-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Elements of Algebraic Forms\"\u003e\u003c/a\u003eElements of Algebraic Forms\u003c/h2\u003e\u003ch3 id=\"variable\" class=\"headerLink\"\u003e\n \u003ca href=\"#variable\" class=\"header-mark\" aria-label=\"Direct link to section: Variable\"\u003e\u003c/a\u003eVariable\u003c/h3\u003e\u003cp\u003eA variable is a symbol used in place of a number whose value is not yet known. In the algebraic form \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x + y - 10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the letters \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are variables.\u003c/p\u003e\n\u003ch3 id=\"coefficient\" class=\"headerLink\"\u003e\n \u003ca href=\"#coefficient\" class=\"header-mark\" aria-label=\"Direct link to section: Coefficient\"\u003e\u003c/a\u003eCoefficient\u003c/h3\u003e\u003cp\u003eA coefficient is the numerical factor attached to a variable. In \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x - y + 7z + 8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ccode\u003e3\u003c/code\u003e is the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e-1\u003c/code\u003e is the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003e7\u003c/code\u003e is the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ez\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"constant\" class=\"headerLink\"\u003e\n \u003ca href=\"#constant\" class=\"header-mark\" aria-label=\"Direct link to section: Constant\"\u003e\u003c/a\u003eConstant\u003c/h3\u003e\u003cp\u003eA constant is a term with no variable. In \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2 - 3x + 9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the number \u003ccode\u003e9\u003c/code\u003e is the constant term.\u003c/p\u003e\n\u003ch3 id=\"term\" class=\"headerLink\"\u003e\n \u003ca href=\"#term\" class=\"header-mark\" aria-label=\"Direct link to section: Term\"\u003e\u003c/a\u003eTerm\u003c/h3\u003e\u003cp\u003eA term is a part of an algebraic expression separated by addition or subtraction. In \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2 - 5x + 12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the terms are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-5x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"like-terms\" class=\"headerLink\"\u003e\n \u003ca href=\"#like-terms\" class=\"header-mark\" aria-label=\"Direct link to section: Like Terms\"\u003e\u003c/a\u003eLike Terms\u003c/h3\u003e\u003cp\u003eLike terms have the same variables raised to the same powers.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4x^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are like terms\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4x^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-7x^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are also like terms\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"unlike-terms\" class=\"headerLink\"\u003e\n \u003ca href=\"#unlike-terms\" class=\"header-mark\" aria-label=\"Direct link to section: Unlike Terms\"\u003e\u003c/a\u003eUnlike Terms\u003c/h3\u003e\u003cp\u003eUnlike terms do not have matching variables or exponents. For example, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are unlike terms.\u003c/p\u003e\n\u003ch2 id=\"operations-on-algebraic-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#operations-on-algebraic-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Operations on Algebraic Forms\"\u003e\u003c/a\u003eOperations on Algebraic Forms\u003c/h2\u003e\u003ch3 id=\"addition-and-subtraction\" class=\"headerLink\"\u003e\n \u003ca href=\"#addition-and-subtraction\" class=\"header-mark\" aria-label=\"Direct link to section: Addition and Subtraction\"\u003e\u003c/a\u003eAddition and Subtraction\u003c/h3\u003e\u003cp\u003eAddition and subtraction can only be done directly on like terms.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x + 3x = 5x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e8a - 6a = 2a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eAdd \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex + 7y - 4z\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-12x + 3y + 10z\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n (x + 7y - 4z) + (-12x + 3y + 10z)\n \u0026amp;= (x - 12x) + (7y + 3y) + (-4z + 10z)\\\\\n \u0026amp;= -11x + 10y + 6z\n \\end{aligned}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSubtract \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6x^2 - 2x + 8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4x^2 - 11x + 12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable 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stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n (4x^2 - 11x + 12) - (6x^2 - 2x + 8)\n \u0026amp;= 4x^2 - 11x + 12 - 6x^2 + 2x - 8\\\\\n \u0026amp;= (4x^2 - 6x^2) + (-11x + 2x) + (12 - 8)\\\\\n \u0026amp;= -2x^2 - 9x + 4\n \\end{aligned}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5723em;vertical-align:-2.0362em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan 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style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan 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mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"multiplication-and-division\" class=\"headerLink\"\u003e\n \u003ca href=\"#multiplication-and-division\" class=\"header-mark\" aria-label=\"Direct link to section: Multiplication and Division\"\u003e\u003c/a\u003eMultiplication and Division\u003c/h3\u003e\u003cp\u003eUseful properties:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nab \u0026amp;= ba\\\\\np(a+b) \u0026amp;= pa + pb\\\\\np(a-b) \u0026amp;= pa - pb\\\\\np(a+b-c) \u0026amp;= pa + pb - pc\\\\\n\\frac{a-b+c-d}{p} \u0026amp;= \\frac{a}{p} - \\frac{b}{p} + \\frac{c}{p} - \\frac{d}{p}, \\quad p \\ne 0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:8.5519em;vertical-align:-4.0259em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.5259em;\"\u003e\u003cspan style=\"top:-7.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eab\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.5259em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0259em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.5259em;\"\u003e\u003cspan style=\"top:-7.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eba\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.5259em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0259em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5(2x+y) = 10x + 5y\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex(4x^2+9x+2) = 4x^3 + 9x^2 + 2x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" 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mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(8x^2-4x):2x = 4x - 2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(12xy^2+6x^2y):3xy = 4y + 2x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"evaluating-algebraic-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluating-algebraic-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluating Algebraic Forms\"\u003e\u003c/a\u003eEvaluating Algebraic Forms\u003c/h2\u003e\u003cp\u003eIf variables are replaced with specific values, the value of the algebraic expression can be calculated.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eFind the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x+4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e when \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e19\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n 3x+4 = 3(5)+4 = 19\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e19\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThe product of two numbers is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e36\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e36\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. If one of the numbers is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, find the other number.\u003c/p\u003e\n\u003cp\u003eLet the unknown number be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e36\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n -4x \u0026amp;= 36\\\\\n x \u0026amp;= \\frac{36}{-4}\\\\\n x \u0026amp;= -9\n \\end{aligned}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3908em;vertical-align:-2.4454em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9454em;\"\u003e\u003cspan style=\"top:-5.4268em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4454em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5361em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4454em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9454em;\"\u003e\u003cspan style=\"top:-5.4268em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4454em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5361em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4454em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the other number is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"powers-in-algebraic-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#powers-in-algebraic-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Powers in Algebraic Forms\"\u003e\u003c/a\u003ePowers in Algebraic Forms\u003c/h2\u003e\u003cp\u003eAlgebraic expressions may also contain powers, such as:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3a^2b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^3 - 5x^2 + x - 7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eAn exponent shows how many times a variable is multiplied by itself. For example:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^3 = x \\times x \\times x\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eUnderstanding variables, coefficients, constants, terms, and the basic arithmetic operations on algebraic expressions is an important foundation for later mathematics topics.\u003c/p\u003e\n", "language": "en" }, { "title" : "Drawing a Cube and Its Elements in LaTeX", "date_published" : "2021-10-04T06:10:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menggambar-kubus-dan-unsur-unsurnya-di/", "url" : "https://www.aantriono.com/en/menggambar-kubus-dan-unsur-unsurnya-di/", "summary": " IntroductionCube shapes are often found around us. Cube-shaped objects include dice, rubik\u0026rsquo;s cubes, and cardboard. Some of the elements contained in the cube are as follows:\nA cube has 6 square sides. A cube has 8 vertices. A cube has 12 edges of the same length. A cube has 12 diagonal planes A cube has 4 diagonal spaces A cube has 6 diagonal faces In this article, I will share a tutorial on how to draw a cube and its elements in LaTeX\\LaTeXLATE​X.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eCube shapes are often found around us. Cube-shaped objects include dice, rubik\u0026rsquo;s cubes, and cardboard. Some of the elements contained in the cube are as follows:\u003c/p\u003e\n\u003col class=\"style1\"\u003e \u003cli\u003eA cube has 6 square sides.\u003c/li\u003e\n\u003cli\u003eA cube has 8 vertices.\u003c/li\u003e\n\u003cli\u003eA cube has 12 edges of the same length.\u003c/li\u003e\n\u003cli\u003eA cube has 12 diagonal planes\u003c/li\u003e\n\u003cli\u003eA cube has 4 diagonal spaces\u003c/li\u003e\n\u003cli\u003eA cube has 6 diagonal faces\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eIn this article, I will share a tutorial on how to draw a cube and its elements in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch2 id=\"preamble\" class=\"headerLink\"\u003e\n \u003ca href=\"#preamble\" class=\"header-mark\" aria-label=\"Direct link to section: Preamble\"\u003e\u003c/a\u003ePreamble\u003c/h2\u003e\u003cp\u003eIn the preamble we will write the document class and packages used to draw the cube elements.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\documentclass{article}\\usepackage{fancybox}\\usepackage{tikz}\\usepackage{enumitem}\u003c/code\u003e\u003c/pre\u003e\n\u003ch2 id=\"document-body\" class=\"headerLink\"\u003e\n \u003ca href=\"#document-body\" class=\"header-mark\" aria-label=\"Direct link to section: Document Body\"\u003e\u003c/a\u003eDocument Body\u003c/h2\u003e\u003cp\u003eIn the body of the document we start writing commands for drawing which of course start with \\begin{tikzpicture} and end with \\end{tikzpictue}. Pay attention to the following code command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{document}\\begin{tikzpicture}[thick,scale=5]\\end{tikzpicture}\\end{document} \u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"drawing-corner-points\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-corner-points\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Corner Points\"\u003e\u003c/a\u003eDrawing Corner Points\u003c/h3\u003e\u003cp\u003eHere, the cube to be drawn is a cube \u003cem\u003eABCD EFGH\u003c/em\u003e. The code is as follows.\u003c/p\u003e\n\u003cpre\u003e \u003ccode\u003e\\coordinate [label=below:{\\color{red}$A$}](A) at (0, 0); \\coordinate [label=below:{\\color{red}$B$}](B) at (1, 0); \\coordinate [label=right:{\\color{red}$C$}](C) at (1.3, 0.3); \\coordinate [label=left:{\\color{red}$D$}](D) at (0.3, 0.3); \\coordinate [label=left:{\\color{red}$E$}](E) at (0, 1); \\coordinate [label=right:{\\color{red}$F$}](F) at (1, 1); \\coordinate [label=above:{\\color{red}$G$}](G) at (1.3, 1.3); \\coordinate [label=above:{\\color{red}$H$}](H) at (0.3, 1.3)\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"drawing-cube-side-symbols\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-cube-side-symbols\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Cube Side Symbols\"\u003e\u003c/a\u003eDrawing Cube Side Symbols\u003c/h3\u003e\u003cp\u003eCube side symbols generally use the lowercase S symbol (s).\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\coordinate[label=below:$s$](s) at (0.5, 0); \\coordinate[label=right:$s$] (s) at (1.15, 0.15); \\coordinate[label=right:$s$] (s) at (1.3, 0.8);\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"drawing-ribs\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-ribs\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Ribs\"\u003e\u003c/a\u003eDrawing Ribs\u003c/h3\u003e\u003cp\u003eNext we will draw the edges of the cube *ABCD EFGH. *Ribs This will connect the corner points on the cube.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e \\draw[blue, very thick] (A) -- (E); \\draw[blue, very thick] (E) -- (F); \\draw[blue, very thick] (B) -- (F); \\draw[blue, very thick] (A) -- (B); \\draw[blue, dashed] (A) -- (D); \\draw[blue, dashed] (D) -- (H); \\draw[blue, very thick] (E) -- (H); \\draw[blue, very thick] (G) -- (H); \\draw[blue, very thick] (F) -- (G); \\draw[blue, very thick] (B) -- (C); \\draw[blue, very thick] (C) -- (G); \\draw[blue, dashed] (C) -- (D);\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eAt this point, if we compile the document script, the results will look like the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwnxl1AsKuW12OCd2X_kuolcmX9cNOx16q-v3pj_zMPQhQHXOixaTDCiNaHfqjgZP2sJpCjKu2eL1tSyVHKZbnp_1aU-PAQwxXZQNt6-EDZ_nP_nPVygksZGcn7gBfALYY6sB-3V_PtHE/s320/KUBUS.jpeg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Kubus dan unsur-unsurnya di LaTeX\" border=\"0\" data-original-height=\"293\" data-original-width=\"320\" height=\"293\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwnxl1AsKuW12OCd2X_kuolcmX9cNOx16q-v3pj_zMPQhQHXOixaTDCiNaHfqjgZP2sJpCjKu2eL1tSyVHKZbnp_1aU-PAQwxXZQNt6-EDZ_nP_nPVygksZGcn7gBfALYY6sB-3V_PtHE/w320-h293/KUBUS.jpeg\" title=\"Kubus dan unsur-unsurnya\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch3 id=\"drawing-planeside-diagonals\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-planeside-diagonals\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Plane/Side Diagonals\"\u003e\u003c/a\u003eDrawing Plane/Side Diagonals\u003c/h3\u003e\u003cp\u003eDiagonal planes/sides of a cube *ABCD EFGH *there are 12, namely: \u003cem\u003eAC, BD, EG, F H, AH, ED, BG, CF, AF, BE, CH, DG\u003c/em\u003e. The command to draw it can be seen in the following code.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e \\draw[red, dashed] (A) -- (C); \\draw[red, dashed] (B) -- (D); \\draw[red, dashed] (E) -- (G); \\draw[red, dashed] (F) -- (H); \\draw[red, dashed] (A) -- (H); \\draw[red, dashed] (E) -- (D); \\draw[red, dashed] (B) -- (G); \\draw[red, dashed] (C) -- (F); \\draw[red, dashed] (A) -- (F); \\draw[red, dashed] (B) -- (E); \\draw[red, dashed] (C) -- (H); \\draw[red, dashed] (D) -- (G);\u003c/code\u003e\u003c/pre\u003e\n\u003ch3 id=\"drawing-space-diagonals\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-space-diagonals\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Space Diagonals\"\u003e\u003c/a\u003eDrawing Space Diagonals\u003c/h3\u003e\u003cp\u003eThe diagonal of the space to be drawn is diagonal AG, BH, CE, DA DF. \\draw[green, dashed] (A) \u0026ndash; (G); \\draw[green, dashed] (B) \u0026ndash; (H); \\draw[green, dashed] (C) \u0026ndash; (E); \\draw[green, dashed] (D) \u0026ndash; (F);\u003c/p\u003e\n\u003ch3 id=\"drawing-diagonal-planes\" class=\"headerLink\"\u003e\n \u003ca href=\"#drawing-diagonal-planes\" class=\"header-mark\" aria-label=\"Direct link to section: Drawing Diagonal Planes\"\u003e\u003c/a\u003eDrawing Diagonal Planes\u003c/h3\u003e\u003cp\u003eDiagonal plane of a cube \u003cem\u003eABCD EFGH\u003c/em\u003e there are 6 pieces, namely:* ACGE, BDHF, ABGH, CDEF, ADGF, BCHE*.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e \\draw[dashed, fill=green, opacity=0.5] (B) -- (C) -- (H) --(E); \\draw[dashed, fill=orange, opacity=0.5] (A) -- (B) -- (G) --(H); \\draw[dashed, fill=blue, opacity=0.5] (A) -- (C) -- (G) --(E); \\draw[dashed, fill=blue, opacity=0.5] (B) -- (D) -- (H) --(F); \\draw[dashed, fill=blue, opacity=0.5] (C) -- (D) -- (E) --(F); \\draw[dashed, fill=blue, opacity=0.5] (A) -- (D) -- (G) --(F);\u003c/code\u003e\u003c/pre\u003e\n\u003ch2 id=\"complete-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete Code\"\u003e\u003c/a\u003eComplete Code\u003c/h2\u003e\u003cp\u003eFor the complete code, I deliberately added several command codes as information about the cube image.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\documentclass{article}\\usepackage{fancybox}\\usepackage{tikz}\\usepackage{enumitem}\\title{Unsur-Unsur Kubus}\\author{Aan Triono}\\date{\\today}\\parindent0em\\begin{document}\\maketitle\\centering\\textcolor{blue!75}{\\shadowbox{KUBUS}}\\vspace{.3 cm}\\begin{tikzpicture}[thick,scale=5]% ~~~~~~~~~~~ Menggambar titik sudut ~~~~~~~~~~~ \\coordinate [label=below:{\\color{red}$A$}](A) at (0, 0); \\coordinate [label=below:{\\color{red}$B$}](B) at (1, 0); \\coordinate [label=right:{\\color{red}$C$}](C) at (1.3, 0.3); \\coordinate [label=left:{\\color{red}$D$}](D) at (0.3, 0.3); \\coordinate [label=left:{\\color{red}$E$}](E) at (0, 1); \\coordinate [label=right:{\\color{red}$F$}](F) at (1, 1); \\coordinate [label=above:{\\color{red}$G$}](G) at (1.3, 1.3); \\coordinate [label=above:{\\color{red}$H$}](H) at (0.3, 1.3); % ~~~~~~~~~~~~~~~ Menggambar Simbol sisi ~~~~~~~~~~~~~~~~~ \\coordinate[label=below:$s$](s) at (0.5, 0); \\coordinate[label=right:$s$] (s) at (1.15, 0.15); \\coordinate[label=right:$s$] (s) at (1.3, 0.8); %\\coordinate[label=below:$.$] (Z) at (1, 0.47); %---------------Rusuk---------------- \\draw[blue, very thick] (A) -- (E); \\draw[blue, very thick] (E) -- (F); \\draw[blue, very thick] (B) -- (F); \\draw[blue, very thick] (A) -- (B); \\draw[blue, dashed] (A) -- (D); \\draw[blue, dashed] (D) -- (H); \\draw[blue, very thick] (E) -- (H); \\draw[blue, very thick] (G) -- (H); \\draw[blue, very thick] (F) -- (G); \\draw[blue, very thick] (B) -- (C); \\draw[blue, very thick] (C) -- (G); \\draw[blue, dashed] (C) -- (D); %~~~~~~~~~~~~ Diagonal Bidang/Sisi ~~~~~~~~~~~ \\draw[red, dashed] (A) -- (C); \\draw[red, dashed] (B) -- (D); \\draw[red, dashed] (E) -- (G); \\draw[red, dashed] (F) -- (H); \\draw[red, dashed] (A) -- (H); \\draw[red, dashed] (E) -- (D); \\draw[red, dashed] (B) -- (G); \\draw[red, dashed] (C) -- (F); \\draw[red, dashed] (A) -- (F); \\draw[red, dashed] (B) -- (E); \\draw[red, dashed] (C) -- (H); \\draw[red, dashed] (D) -- (G); %~~~~~~~~~~ Diagonal Ruang ~~~~~~~~~~~~ \\draw[green, dashed] (A) -- (G); \\draw[green, dashed] (B) -- (H); \\draw[green, dashed] (C) -- (E); \\draw[green, dashed] (D) -- (F); % ~~~~~~~~~~~ Bidang Diagonal~~~~~~~~~~~~~~~ \\draw[dashed, fill=green, opacity=0.5] (B) -- (C) -- (H) --(E); \\draw[dashed, fill=orange, opacity=0.5] (A) -- (B) -- (G) --(H); %\\draw[dashed, fill=blue, opacity=0.5] (A) -- (C) -- (G) --(E); %\\draw[dashed, fill=blue, opacity=0.5] (B) -- (D) -- (H) --(F); %\\draw[dashed, fill=blue, opacity=0.5] (C) -- (D) -- (E) --(F); %\\draw[dashed, fill=blue, opacity=0.5] (A) -- (D) -- (G) --(F); \\end{tikzpicture}\\vspace{0.5cm}Unsur-unsur kubus :\\begin{itemize} \\item Bidang : $ABCD, EFGH, ADHE, BCGF, DCGH.$ \\item Rusuk : $\\overline{AB}, \\overline{BC}, \\overline{CD}, \\overline{AD}, \\overline{EF}, \\overline{FG}, \\overline{GH}, \\overline{EH}, \\overline{AE}, \\overline{BF}, \\overline{CG}, \\overline{DH}.$ \\item Titik sudut : $A, B, C. D, F, G, H.$ \\item Diagonal sisi : $AC, BD, EG, FH, AH, ED, BG, CF, AF, BE, CH, DG.$ \\item Diagonal ruang : $AG, BH, CE, DF.$ \\item Bidang diagonal : $ACGE, BDHF, ADGF, BCHE, CDEF, ABGH.$\\end{itemize}\\fcolorbox{red}{blue!40}{Luas permukaan kubus, $L=6s^2$} dan \\fcolorbox{red}{blue!40}{Volume kubus, $V=s^3$}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eYou can compile these codes and see the results using the LaTeX engine on this blog. Please try \u003ca href=\"https://www.silagebra.com/p/latex-engine.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e. The results of compiling the document script will look like the image in the following document.\u003c/p\u003e\n\u003ciframe allow=\"autoplay\" height=\"480\" src=\"https://drive.google.com/file/d/1IEtATzGCwC3v37awPbxHse_7-3mQseEr/preview\" width=\"640\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat\u0026rsquo;s how to draw a cube and its elements in LaTeX. Of course, not all elements are written or described here. You can develop it more fully. If there are errors, please correct them, and if you have input, please leave them in the comments. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Class 9 Odd Semester Assessment Quiz", "date_published" : "2021-09-30T15:42:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kuis-interaktif-penilaian-tengah/", "url" : "https://www.aantriono.com/en/kuis-interaktif-penilaian-tengah/", "summary": " IntroductionOn this occasion, I made a post about the odd mid-semester assessment quiz for class 9 SMP/MTs with multiple choice questions, where the question script had been previously created using LaTeX. One of the advantages of writing question papers in LaTeX is that the display of mathematical formulas or symbols is neat and in accordance with scientific rules for writing mathematical notation. To see a guide to making mid-semester assessment questions using LaTeX, you can read the post Creating Midterm Assessment Questions in LaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eOn this occasion, I made a post about the odd mid-semester assessment quiz for class 9 SMP/MTs with multiple choice questions, where the question script had been previously created using LaTeX. One of the advantages of writing question papers in LaTeX is that the display of mathematical formulas or symbols is neat and in accordance with scientific rules for writing mathematical notation. To see a guide to making mid-semester assessment questions using LaTeX, you can read the post \u003ca href=\"https://www.silagebra.com/2021/09/membuat-soal-penilaian-tengah-semester.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eCreating Midterm Assessment Questions in LaTeX\u003c/a\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIgt5w4OHSiJEMGjE6JIYaneBrZr4la6KHuLb6b9-nJ7dKkkj9wspmiZq-BGMJXXoPeGX3TaKISlpYunOBf5qExbgB5_t2PkU3LwJBnV0drqb2hzjsSJsJJmtT9oB3_MCPJbMJZaYnBDA/s1280/photo1633016016.jpeg' alt=\"Odd mid-semester assessment class 9\" \u003e\u003c/p\u003e\n\u003ch2 id=\"quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Quiz\"\u003e\u003c/a\u003eQuiz\u003c/h2\u003e\u003cp\u003eFurthermore, after completing the questions, students can work on the questions directly on this blog with the note that what is presented here is specifically for 10 multiple choice questions.\u003c/p\u003e\n\u003ch2 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also Read\"\u003e\u003c/a\u003eAlso Read\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-bentuk-akar.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eRoot Form Material Quiz\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-bilangan-berpangkat.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eRanked Numbers Material Quiz\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-persamaan-kuadrat.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ePower Equation Material Quiz\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"instructions-for-taking-the-quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#instructions-for-taking-the-quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Instructions for Taking the Quiz\"\u003e\u003c/a\u003eInstructions for Taking the Quiz\u003c/h3\u003e\u003col class=\"style1\"\u003e\n \u003cli\u003eClick the button Start to start the quiz.\u003c/li\u003e\n \u003cli\u003eWrite Name And Class you in the box provided.\u003c/li\u003e\n \u003cli\u003eClick the button Do it to go to the question menu.\u003c/li\u003e\n \u003cli\u003eClick on the button which is the correct answer.\u003c/li\u003e\n \u003cli\u003eClick on the sign \u003e to go to the next question, and click the sign \u003c to return to the previous question, or click question menu to see the whole question.\u003c/li\u003e\n \u003cli\u003eThe quiz time is: 30 minutes.\u003c/li\u003e\n \u003cli\u003eThe KKM for this quiz is 70\u003c/li\u003e\n \u003cli\u003eClick the button Finished to end the quiz.\u003c/li\u003e\n \u003cli\u003eMark You will appear after clicking the finish button.\u003c/li\u003e\n \u003c/ol\u003e\n\u003c!--Tombol Mulai--\u003e\n\u003cdiv class=\"center\"\u003e\n \u003cbutton id=\"klik-mulai\"\u003eStart\u003c/button\u003e\n\u003c!--Box Input Nama dan Kelas--\u003e\n\u003cdiv id=\"input-data\"\u003e\n\u003c!--Box Soal--\u003e\n\u003cdiv id=\"bank-soal\"\u003e\n\u003cdiv class=\"waktu-wrap\"\u003e\n\u003cdiv class=\"data-peserta\"\u003e\n\u003cdiv class=\"nama-peserta\"\u003e\n\u003cdiv class=\"kelas-peserta\"\u003e\n\u003c!--Waktu Soal, Ganti berdasarkan jam--\u003e\n\u003cdiv data-waktu=\"0.5\" id=\"waktu-soal\"\u003e\n \u003c!--Jumlah Soal, Ganti berdasrkan jumlah soal--\u003e\n\u003cdiv class=\"bank-soal-wrap\" data-soal=\"10\"\u003e\n\u003c!--Pertanyaan dan Jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e1. The value of $8^\\frac{4}{3}+9^\\frac{3}{2}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$23$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$33$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$43$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$53$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Penutup kode pertanyaan dan jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e2. The result of $1,000^0+100^0+10^0+1^0+\\bigg(\\dfrac{1}{10}\\bigg)^0=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1,112$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1,111$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$5$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e3. The result of $(-8m^2 n^3 ) \\times (2k^3 n^4 )=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-16k^3 m^2 n^{12}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-16k^3 m^2 n^7$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$16k^3 m^2 n^7$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$16k^3 m^2 n^7$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e4. The result of $\\dfrac{4^\\frac{-2}{3}\\times 4^\\frac{-4}{3}}{4^{-3}}=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$4$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e5. If $\\dfrac{64}{729}=(\\dfrac{2}{3})^{x-2}$ then the value of $x^2+1=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$65$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$55$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$45$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$35$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e6. The simple form of $\\sqrt{8}+\\sqrt{32}+\\sqrt{72}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$12\\sqrt{2}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$12\\sqrt{3}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$12\\sqrt{5}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$12\\sqrt{7}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e7. The simple form of $\\sqrt{p}+\\sqrt{4p}-\\sqrt{9p}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$0$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e8. The simple form of $\\dfrac{\\sqrt{6}\\times \\sqrt{2}}{\\sqrt{3}}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$0$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e9. The rational form of $\\dfrac{3}{\\sqrt{2}}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$\\dfrac{3\\sqrt{2}}{2}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3\\sqrt{2}}{3}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3\\sqrt{3}}{2}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3\\sqrt{2}}{3}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e10. The rational form of $\\sqrt{0,1}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{4}{10}\\sqrt{10}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3}{10}\\sqrt{10}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{2}{10}\\sqrt{10}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$\\dfrac{1}{10}\\sqrt{10}$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Tombol Soal, Sesuaikan dengan jumlah soal--\u003e\n\u003cdiv id=\"show-hide-box-soal\"\u003e\n\u003cdiv class=\"tombol-show-hide lihat-soal\"\u003e\n closed\n\u003cdiv class=\"menu-tombol-soal\"\u003e\n\u003cdiv class=\"tombol-soal\" data-nomor=\"1\"\u003e1\n\u003cdiv class=\"tombol-soal\" data-nomor=\"2\"\u003e2\n\u003cdiv class=\"tombol-soal\" data-nomor=\"3\"\u003e3\n\u003cdiv class=\"tombol-soal\" data-nomor=\"4\"\u003e4\n\u003cdiv class=\"tombol-soal\" data-nomor=\"5\"\u003e5\n\u003cdiv class=\"tombol-soal\" data-nomor=\"6\"\u003e6\n\u003cdiv class=\"tombol-soal\" data-nomor=\"7\"\u003e7\n\u003cdiv class=\"tombol-soal\" data-nomor=\"8\"\u003e8\n\u003cdiv class=\"tombol-soal\" data-nomor=\"9\"\u003e9\n\u003cdiv class=\"tombol-soal\" data-nomor=\"10\"\u003e10\n\u003cdiv id=\"menu-soal\"\u003e\n\u003cdiv class=\"next-prev-soal\"\u003e\n\u003cdiv class=\"prev\" id=\"prev-soal\"\u003e\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M15.41,16.58L10.83,12L15.41,7.41L14,6L8,12L14,18L15.41,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n Previously\n\u003cdiv class=\"next\" id=\"next-soal\"\u003e\n Furthermore\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M8.59,16.58L13.17,12L8.59,7.41L10,6L16,12L10,18L8.59,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n\u003cdiv class=\"menu-pilihan-soal\"\u003e\n \u003cbutton class=\"lihat-soal\" id=\"lihat-soal\"\u003eQuestion Menu\u003c/button\u003e\n\u003cdiv class=\"tombol-soal-selesai\"\u003e\n \u003cbutton id=\"soal-selesai\"\u003eFinished\u003c/button\u003e\n\u003cdiv id=\"lihat-hasil-nilai\"\u003e\n\u003cdiv id=\"hasil-pengerjaan\"\u003e\n\u003cp class=\"info-keterangan-hasil\"\u003e*Send the results via WhatsApp or Telegram at \u003ca href=\"https://www.aantriono.com/p/contact.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\n\u003cp\u003e, if remedial, you can repeat it again by clicking \u003ca href=\"https://www.silagebra.com/2021/09/membuat-soal-penilaian-tengah-semester.html\"\u003estart\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e.\u003c/p\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat\u0026rsquo;s the article about the odd mid-semester assessment quiz for class 9 SMP/MTs with multiple choice questions that I can share. If there are errors, please correct them, and if you have input, please leave them in the comments. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Creating Midterm Assessment Questions in LaTeX", "date_published" : "2021-09-25T15:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/membuat-soal-penilaian-tengah-semester/", "url" : "https://www.aantriono.com/en/membuat-soal-penilaian-tengah-semester/", "summary": "PIn this article, I will share an article about how to create mid-semester assessment questions in LaTeX and create a quiz on the blog. Mid-semester assessment is an activity carried out by schools to measure student achievement after carrying out learning activities for half a semester based on the educational calendar that has been prepared by each school.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003ePIn this article, I will share an article about how to create mid-semester assessment questions in LaTeX and create a quiz on the blog. Mid-semester assessment is an activity carried out by schools to measure student achievement after carrying out learning activities for half a semester based on the educational calendar that has been prepared by each school.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_8JMVmRsxFBMUsh8T5O4yPLI4S_W7AVrDr6Aj70TiRxbBzw98x5TdIejjqTNUefWQSxstdHtvE2vJosxGypatWXadxbt4p2BxP0KuWX9k7QJf-5uu2jngxzXsbOHDDZuaqfMyEwh_hmY/s490/kuis.jpeg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"menulis soal pts di latex\" border=\"0\" data-original-height=\"370\" data-original-width=\"490\" height=\"242\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_8JMVmRsxFBMUsh8T5O4yPLI4S_W7AVrDr6Aj70TiRxbBzw98x5TdIejjqTNUefWQSxstdHtvE2vJosxGypatWXadxbt4p2BxP0KuWX9k7QJf-5uu2jngxzXsbOHDDZuaqfMyEwh_hmY/w320-h242/kuis.jpeg\" title=\"menulis soal pts di latex\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"midterm-assessment-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#midterm-assessment-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Midterm Assessment Objectives\"\u003e\u003c/a\u003eMidterm Assessment Objectives\u003c/h2\u003e\u003cp\u003eThe objectives of holding the mid-semester assessment include:\u003c/p\u003e\n\u003col class=\"style1\"\u003e \u003cli\u003eMeasuring student learning progress after half the semester.\u003c/li\u003e\n\u003cli\u003eMeasuring student competency achievement after a half-semester learning process. \u003c/li\u003e\n\u003cli\u003eDetermining the value of student learning outcomes after the learning process of several KD (Basic Competencies).\u003c/li\u003e\n\u003cli\u003eMake learning improvements in the next half semester.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"writing-midterm-assessment-questions-in-latex\" class=\"headerLink\"\u003e\n \u003ca href=\"#writing-midterm-assessment-questions-in-latex\" class=\"header-mark\" aria-label=\"Direct link to section: Writing Midterm Assessment Questions in LaTeX\"\u003e\u003c/a\u003eWriting Midterm Assessment Questions in LaTeX\u003c/h2\u003e\u003cp\u003eHere, I will write LaTeX codes for creating mid-semester assessment questions for classes IX and VII. In the document class and preamble section, I use the following command. The multicol package is used to divide the manuscript into 2 columns.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\documentclass[legal, 11pt]{article}\\usepackage[indonesian]{babel}\\usepackage{fourier}\\usepackage{xcolor}\\usepackage[margin=1cm]{geometry}\\usepackage{array, multicol}\\usepackage{amsmath}\\usepackage[shortlabels]{enumitem}\\usepackage{graphicx}\\usepackage{adjustbox}\\usepackage{calligra}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eNext, in the body of the document, I use the following command to place the district logo and school name.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\noindent\\adjustbox{valign=c}{\\begin{minipage}{0.1\\textwidth}\\includegraphics[width=3.5cm, height=3.5cm\t\t\t]{logolt}\\end{minipage}}\\hfill\\adjustbox{valign=c}{\\begin{minipage}{0.8\\textwidth}\\begin{center}\\textbf{\t\\large {PEMERINTAH KABUPATEN LAMPUNG TENGAH}\\\\\t\t\t\tDINAS PENDIDIKAN DAN KEBUDAYAAN\\\\\t\t\t\tUPTD SATUAN PENDIDIKAN SMP NEGERI PURWOREJO\\\\\t\t\t\tKECAMATAN PADANG RATU\\\\\t\t\t\t\\small {Alamat : Jln. Perintis Rogowungu Purworejo Kec. Padang Ratu Kab. Lampung Tengah 34176}\\\\}\\end{center}\t\t\\end{minipage}}\\hfill\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eTo put a double line under the logo use the following command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\hrule\\vspace{.75mm}\\hrule height 1.5pt\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eNext, create the identity of the mid-semester assessment questions using the following command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{center}\\textbf{PENILAIAN TENGAH SEMESTER (PTS)\\\\SEMESTER GANJIL TAHUN PELAJARAN 2021/2022\\\\}\\vspace{.2cm}\\begin{table}[!ht]\\small\\begin{tabular}{lclp{5cm}lcl}\tMata Pelajaran\u0026:\u0026Matematika\u0026\u0026Hari/Tanggal\u0026:\u0026Selasa, 12 Oktober 2021 \\\\\tNama\u0026:\u0026 \u0026\u0026Kelas\u0026:\u0026IX \\\\\tNo. Peserta Ujian\u0026:\u0026 \u0026\u0026Waktu\u0026:\u002607.30-08.30 WIB\\\\\\end{tabular}\\end{table}\\end{center} \u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eNext, write instructions for working on the questions using the following command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{enumerate}[A.]\t\t\\item \\textbf{Petunjuk Khusus Soal Pilihan Ganda}\\\\\t\t%\\begin{enumerate}[1.]Jawablah 10 butir soal pilihan ganda berikut dengan menyilang (X) salah satu opsi jawaban (A, B, C, atau D) pada lembar soal yang telah disediakan dan pastikan hanya ada satu jawaban untuk setiap butir soal serta ketiga pilihan jawaban yang lain dalam keadaan bersih tanpa coretan! \t\\item \\textbf{Petunjuk Khusus Soal Esai}\\\\Jawablah 5 butir soal esai berikut dengan benar dan pada tempat yang disediakan.\t\\end{enumerate}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThen start to write multiple choice questions which will be divided into 2 columns using the following command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{multicols}{2}\t\t\t\\begin{enumerate}[1.,leftmargin=.5cm,itemsep=0pt,topsep=1pt]\t\t\t\t\\item Nilai dari $8^\\frac{4}{3}+9^\\frac{3}{2}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $23$\\item $33$\\item $43$\\item $53$\\end{enumerate}\t\t\t\t\\item Hasil dari $1.000^0+100^0+10^0+1^0+(\\dfrac{1}{10})^0=\\cdots$\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $1.112$\\item $1.111$\\item $5$\\item $1$\\end{enumerate}\t\t\t\t\\item Hasil dari $(-8m^2 n^3 ) \\times (2k^3 n^4 )=\\cdots$\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $-16k^3 m^2 n^{12}$\\item $-16k^3 m^2 n^7$\\item $16k^3 m^2 n^7$\\item $16k^3 m^2 n^7$\\end{enumerate} \\item Hasil dari $\\dfrac{4^\\frac{-2}{3}\\times 4^\\frac{-4}{3}}{4^{-3}}=\\cdots$ \\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $1$\\item $2$\\item $3$\\item $4$\\\\[1pt]\\end{enumerate}\t\t\t\t\\item Jika $\\dfrac{64}{729}=(\\dfrac{2}{3})^{x-2}$ maka nilai $x^2+1=\\cdots$\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $65$\\item $55$\\item $45$\\item $35$\\end{enumerate}\t\t\t\t\\item Bentuk sederhana dari $\\sqrt{8}+\\sqrt{32}+\\sqrt{72}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $12\\sqrt{2}$\\item $12\\sqrt{3}$\\item $12\\sqrt{5}$\\item $12\\sqrt{7}$\\end{enumerate}\t\t\t\t\\item Bentuk sederhana dari $\\sqrt{p}+\\sqrt{4p}-\\sqrt{9p}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $3$\\item $2$\\item $1$\\item $0$\\end{enumerate}\t\t\t\t\\item Bentuk sederhana dari $\\dfrac{\\sqrt{6}\\times \\sqrt{2}}{\\sqrt{3}}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $0$\\item $1$\\item $2$\\item $3$\\end{enumerate}\t\t\t\t\\item Bentuk rasional dari $\\dfrac{3}{\\sqrt{2}}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $\\dfrac{3\\sqrt{2}}{2}$\\\\[3pt]\\item $\\dfrac{3\\sqrt{2}}{3}$\\\\[3pt]\\item $\\dfrac{3\\sqrt{3}}{2}$\\\\[3pt]\\item $\\dfrac{3\\sqrt{2}}{3}$\\\\[3pt]\\end{enumerate}\t\t\t\t\\item Bentuk rasional dari $\\sqrt{0,1}$ adalah ...\t\t\t\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $\\dfrac{4}{10}\\sqrt{10}$\\\\[3pt]\\item $\\dfrac{3}{10}\\sqrt{10}$\\\\[3pt]\\item $\\dfrac{2}{10}\\sqrt{10}$\\\\[3pt]\\item $\\dfrac{1}{10}\\sqrt{10}$\\\\[3pt]\\end{enumerate}\t\t\t\\end{enumerate}\t\t\\end{multicols}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eFurthermore, in the essay questions section there are no longer 2 columns, for example I make questions like the following.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\begin{enumerate}[1.,leftmargin=1.4cm,itemsep=0pt,topsep=1pt]\t \\item Persamaan $x^2-1=2(x-3)$ memiliki bentuk umum $ax^2+bx+c=0$. Nilai $a+b+c=\\cdots$\t\t\\item Himpunan penyelesaian dari persamaan $x^2+7x+12=0$ adalah ...\t\t\\item Jenis akar-akar dari persamaan $x^2-6x+9=0$ adalah ...\t\t\\item Akar-akar persamaan $x^2-12=4x$ adalah $x_1$ dan $x_2$. Nilai dari $x_1+x_2=\\cdots$\t\t\\item Jika $x=2$ merupakan salah satu akar persamaan $x^2+5x+c=0$, maka nilai $c=\\cdots$\t\\end{enumerate}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eAnd at the bottom of the question, I wrote a letter to encourage students, for example good luck, through the following command.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\vspace{18cm}\\begin{center}{\\LARGE\\calligra\\color{cyan!80!orange} Semoga Sukses}\\end{center}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eNext, repeat the steps above to rewrite the LaTeX code to create class VII midterm assessment questions.\u003c/p\u003e\n\u003ch2 id=\"complete-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#complete-code\" class=\"header-mark\" aria-label=\"Direct link to section: Complete Code\"\u003e\u003c/a\u003eComplete Code\u003c/h2\u003e\u003cp\u003eBelow is the complete \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e code for the document.\u003c/p\u003e\n\u003cpre\u003e\u003ccode\u003e\\documentclass[legal, 11pt]{article}\\usepackage[indonesian]{babel}\\usepackage{fourier}\\usepackage{xcolor}\\usepackage[margin=1cm]{geometry}\\usepackage{array, multicol}\\usepackage{amsmath}\\usepackage[shortlabels]{enumitem}\\usepackage{graphicx}\\usepackage{adjustbox}\\usepackage{calligra}\\let\\ds\\displaystyle\\begin{document}\\thispagestyle{empty}\t\\noindent\\adjustbox{valign=c}{\\begin{minipage}{0.1\\textwidth}\\includegraphics[width=3.5cm, height=3.5cm\t\t\t]{logolt}\t\\end{minipage}}\\hfill\\adjustbox{valign=c}{\\begin{minipage}{0.8\\textwidth}\t\t\\begin{center}\t\t\t\\textbf{\t\\large {PEMERINTAH KABUPATEN LAMPUNG TENGAH}\\\\\t\t\t\tDINAS PENDIDIKAN DAN KEBUDAYAAN\\\\\t\t\t\tUPTD SATUAN PENDIDIKAN SMP NEGERI PURWOREJO\\\\\t\t\t\tKECAMATAN PADANG RATU\\\\\t\t\t\t\\small {Alamat : Jln. Perintis Rogowungu Purworejo Kec. Padang Ratu Kab. Lampung Tengah 34176}\\\\}\t\t\\end{center}\t\t\\end{minipage}}\\hfil\\vspace{-3mm}\\hrule\\vspace{.75mm}\\hrule height 1.5pt\\begin{center}\\textbf{PENILAIAN TENGAH SEMESTER (PTS)\\\\SEMESTER GANJIL TAHUN PELAJARAN 2021/2022\\\\}\\vspace{.2cm}\\begin{table}[!ht]\\small\\begin{tabular}{lclp{5cm}lcl}\tMata Pelajaran\u0026:\u0026Matematika\u0026\u0026Hari/Tanggal\u0026:\u0026Selasa, 12 Oktober 2021 \\\\\tNama\u0026:\u0026 \u0026\u0026Kelas\u0026:\u0026IX \\\\\tNo. Peserta Ujian\u0026:\u0026 \u0026\u0026Waktu\u0026:\u002607.30-08.30 WIB\\\\\\end{tabular}\\end{table}\\end{center}\t\\begin{enumerate}[A.]\t\t\\item \\textbf{Petunjuk Khusus Soal Pilihan Ganda}\\\\\t\t%\\begin{enumerate}[1.]Jawablah 10 butir soal pilihan ganda berikut dengan menyilang (X) salah satu opsi jawaban (A, B, C, atau D) pada lembar soal yang telah disediakan dan pastikan hanya ada satu jawaban untuk setiap butir soal serta ketiga pilihan jawaban yang lain dalam keadaan bersih tanpa coretan!\t\t\\begin{multicols}{2}\t\t\t\\begin{enumerate}[1.,leftmargin=.5cm,itemsep=0pt,topsep=1pt]\t\t\t\t\\item Nilai dari $8^\\frac{4}{3}+9^\\frac{3}{2}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $23$\\item $33$\\item $43$\\item $53$\\end{enumerate}\t\t\t\t\\item Hasil dari $1.000^0+100^0+10^0+1^0+\\bigg(\\dfrac{1}{10}\\bigg)^0=\\cdots$\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $1.112$\\item $1.111$\\item $5$\\item $1$\\end{enumerate}\t\t\t\t\\item Hasil dari $(-8m^2 n^3 ) \\times (2k^3 n^4 )=\\cdots$\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $-16k^3 m^2 n^{12}$\\item $-16k^3 m^2 n^7$\\item $16k^3 m^2 n^7$\\item $16k^3 m^2 n^7$\\end{enumerate} \\item Hasil dari $\\dfrac{4^\\frac{-2}{3}\\times 4^\\frac{-4}{3}}{4^{-3}}=\\cdots$ \\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $1$\\item $2$\\item $3$\\item $4$\\\\[1pt]\\end{enumerate}\t\t\t\t\\item Jika $\\dfrac{64}{729}=(\\dfrac{2}{3})^{x-2}$ maka nilai $x^2+1=\\cdots$\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $65$\\item $55$\\item $45$\\item $35$\\end{enumerate}\t\t\t\t\\item Bentuk sederhana dari $\\sqrt{8}+\\sqrt{32}+\\sqrt{72}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $12\\sqrt{2}$\\item $12\\sqrt{3}$\\item $12\\sqrt{5}$\\item $12\\sqrt{7}$\\end{enumerate}\t\t\t\t\\item Bentuk sederhana dari $\\sqrt{p}+\\sqrt{4p}-\\sqrt{9p}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $3$\\item $2$\\item $1$\\item $0$\\end{enumerate}\t\t\t\t\\item Bentuk sederhana dari $\\dfrac{\\sqrt{6}\\times \\sqrt{2}}{\\sqrt{3}}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $0$\\item $1$\\item $2$\\item $3$\\end{enumerate}\t\t\t\t\\item Bentuk rasional dari $\\dfrac{3}{\\sqrt{2}}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $\\dfrac{3\\sqrt{2}}{2}$\\\\[3pt]\\item $\\dfrac{3\\sqrt{2}}{3}$\\\\[3pt]\\item $\\dfrac{3\\sqrt{3}}{2}$\\\\[3pt]\\item $\\dfrac{3\\sqrt{2}}{3}$\\\\[3pt]\\end{enumerate}\t\t\t\t\\item Bentuk rasional dari $\\sqrt{0,1}$ adalah ...\t\t\t\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $\\dfrac{4}{10}\\sqrt{10}$\\\\[3pt]\\item $\\dfrac{3}{10}\\sqrt{10}$\\\\[3pt]\\item $\\dfrac{2}{10}\\sqrt{10}$\\\\[3pt]\\item $\\dfrac{1}{10}\\sqrt{10}$\\\\[3pt]\\end{enumerate}\t\t\t\\end{enumerate}\t\t\\end{multicols}\t\t\\newpage\t\t\\thispagestyle{empty}\t\t\\item \\textbf{Petunjuk Khusus Soal Esai}\\\\Jawablah 5 butir soal esai berikut dengan benar dan pada tempat yang disediakan.\t\\end{enumerate}\t%\\begin{enumerate}[]\t\\begin{enumerate}[1.,leftmargin=1.4cm,itemsep=0pt,topsep=1pt]\t \\item Persamaan $x^2-1=2(x-3)$ memiliki bentuk umum $ax^2+bx+c=0$. Nilai $a+b+c=\\cdots$\t\t\\item Himpunan penyelesaian dari persamaan $x^2+7x+12=0$ adalah ...\t\t\\item Jenis akar-akar dari persamaan $x^2-6x+9=0$ adalah ...\t\t\\item Akar-akar persamaan $x^2-12=4x$ adalah $x_1$ dan $x_2$. Nilai dari $x_1+x_2=\\cdots$\t\t\\item Jika $x=2$ merupakan salah satu akar persamaan $x^2+5x+c=0$, maka nilai $c=\\cdots$\t\\end{enumerate}\t %\\end{enumerate} \\vspace{18cm}\\begin{center}{\\LARGE\\calligra\\color{cyan!80!orange} Semoga Sukses}\\end{center}\\newpage\\thispagestyle{empty}\t\\noindent\\adjustbox{valign=c}{\\begin{minipage}{0.1\\textwidth}\\includegraphics[width=3.5cm, height=3.5cm\t\t\t]{logolt}\t\\end{minipage}}\\hfill\\adjustbox{valign=c}{\\begin{minipage}{0.8\\textwidth}\t\t\\begin{center}\t\t\t\\textbf{\t\\large {PEMERINTAH KABUPATEN LAMPUNG TENGAH}\\\\\t\t\t\tDINAS PENDIDIKAN DAN KEBUDAYAAN\\\\\t\t\t\tUPTD SATUAN PENDIDIKAN SMP NEGERI PURWOREJO\\\\\t\t\t\tKECAMATAN PADANG RATU\\\\\t\t\t\t\\small {Alamat : Jln. Perintis Rogowungu Purworejo Kec. Padang Ratu Kab. Lampung Tengah 34176}\\\\}\t\t\\end{center}\t\t\\end{minipage}}\\hfil\\vspace{-3mm}\\hrule\\vspace{.75mm}\\hrule height 1.5pt\\begin{center}\\textbf{PENILAIAN TENGAH SEMESTER (PTS)\\\\SEMESTER GANJIL TAHUN PELAJARAN 2021/2022\\\\}\\vspace{.2cm}\\begin{table}[!ht]\\small\\begin{tabular}{lclp{4.5cm}lcl}\tMata Pelajaran\u0026:\u0026Matematika\u0026\u0026Hari/Tanggal\u0026:\u0026Selasa, 12 Oktober 2021 \\\\\tNama\u0026:\u0026 \u0026\u0026Kelas\u0026:\u0026VII \\\\\tNo. Peserta Ujian\u0026:\u0026 \u0026\u0026Waktu\u0026:\u002607.30-08.30 WIB\\\\\\end{tabular}\\end{table}\\end{center}\t\\begin{enumerate}[A.]\t\t\\item \\textbf{Petunjuk Khusus Soal Pilihan Ganda}\\\\\t\t%\\begin{enumerate}[1.]Jawablah 10 butir soal pilihan ganda berikut dengan menyilang (X) salah satu opsi jawaban (A, B, C, atau D) pada lembar soal yang telah disediakan dan pastikan hanya ada satu jawaban untuk setiap butir soal serta ketiga pilihan jawaban yang lain dalam keadaan bersih tanpa coretan!\t\t%\\end{enumerate}\t\t\\begin{multicols}{2}\t\t\\begin{enumerate}[1.,leftmargin=.5cm,itemsep=0pt,topsep=1pt]\t\t\t\t \\item Nilai dari 1 - 2 + 3 - 4 + 5 - 6 + ... + 9 - 10 adalah ...\t\t \\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $-5$\\item $-2$\\item $0$\\item $3$\\end{enumerate}\t\t\t\t\\item Jika suhu mula-mula suatu objek $5^0C$, kemudian turun $12^0C$. Maka suhu akhir objek tersebut adalah ... \t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $-4$\\item $-5$\\item $-6$\\item $-7$\\end{enumerate}\t\t\t\t\\item Nilai p yang memenuhi $15 :(p + 1) = -3$ adalah ...\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $-7$\\item $-6$\\item $-5$\\item $-4$\\end{enumerate}\t\t\t\t\\item Seekor bangau yang berada $6$ m di atas permukaan air laut tengah mengintai seekor ikan yang berada pada kedalaman $6$ m di bawah permukaan laut. Jarak bangau dan ikan tersebut adalah ... m\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $-12$\\item $-6$\\item $6$\\item $12$\\end{enumerate} \t\t\t\t\\item Dalam suatu tes, jawaban yang benar diberi nilai $4$, yang salah diberi nilai $-2$, dan untuk soal tidak dijawab diberi nilai $0$. Jika dari $25$ soal, Andika menjawab dengan benar $18$ soal dan $5$ soal salah serta sisanya tidak dijawab, maka nilai yang diperoleh Andika adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $60$\\item $61$\\item $62$\\item $63$\\end{enumerate}\t\t\t\t\\item Ibu membeli $40$ kg gula pasir. Gula itu akan dijual eceran dengan dibungkus plastik masing-masing beratnya $\\dfrac{1}{4}$ kg. Banyak kantong plastik berisi gula yang diperlukan adalah ... kantong.\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $10$\\item $80$\\item $120$\\item $160$\\end{enumerate}\t\t\t\t\\item Hasil dari $\\dfrac{1}{3}\\times \\dfrac{5}{4} - 1\\dfrac{2}{3} + \\dfrac{1}{2}$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $\\dfrac{2}{3}$\\\\[4pt]\\item $\\dfrac{3}{4}$\\\\[4pt]\\item $-\\dfrac{3}{4}$\\\\[4pt]\\item $-\\dfrac{1}{4}$\\end{enumerate}\t\t\t\t\\item KPK dari $90$ dan $168$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $2.520$\\item $2.521$\\item $2.522$\\item $2.523$\\end{enumerate}\t\t\t\t\\item FPB dari bilangan $20$ dan $12$ adalah ...\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $1$\\item $2$\\item $3$\\item $4$\\end{enumerate}\t\t\t\t\\item Nita mengunjungi perpustakaan 3 hari sekali dan Rino 4 hari sekali. Apabila tanggal 1 Juli 2021 mereka mengunjungi perpustakaan bersama-sama, mereka akan mengunjungi perpustakaan kembali secara bersama-sama pada tanggal...\tJuli\t\t\t\t\t\t\\begin{enumerate}[A.,leftmargin=*,itemsep=0pt,topsep=2pt]\\item $13$\\item $14$\\item $15$\\item $16$\\end{enumerate}\t\t\t\\end{enumerate}\t\t\\end{multicols}\t\t%\\newpage\t\t\\thispagestyle{empty}\t\t\\item \\textbf{Petunjuk Khusus Soal Esai}\\\\Jawablah 5 butir soal esai berikut dengan benar dan pada tempat yang disediakan.\t\\end{enumerate}\t%\\begin{enumerate}[]\t\\begin{enumerate}[1.,leftmargin=1.4cm,itemsep=0pt,topsep=1pt]\t \\item Jika $Q = \\{\\text{huruf pembentuk kalimat \"SAHABAT SAYA BAIK SEKALI\"}\\}$, nilai $n(Q) =\\cdots$\t\t\\item Jika $V = \\{a, e, i, o, u\\}$, maka banyaknya himpunan bagian dari V adalah ...\t\t\\item Diketahui $A=\\{2, 4, 6, 8, 10, 12\\}$ dan $B=\\{2, 6, 10, 14, 18\\}$. Tentukan $A\\cup B$\t\t\\item Diketahui $A=\\{1, 2, 3, 4, 5\\}$ dan $B=\\{4, 5, 6, 7\\}$. Tentukan $A\\cap B$\t\t\\item Jumlah siswa kelas VIIA 32 orang, 20 orang diantaranya menyukai pelajaran matematika dan 14 orang menyukai bahasa Inggris. Berapa siswa yang menyukai pelajaran matematika dan bahasa Inggris?\t\\end{enumerate}\t %\\end{enumerate} \\vspace{15cm}\\begin{center}{\\LARGE\\calligra\\color{cyan} Semoga Berhasil}\\end{center}\\end{document}\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003eThe final result of the document script that has been created will look like the following document.\u003c/p\u003e\n\u003ciframe allow=\"autoplay\" height=\"480\" src=\"https://drive.google.com/file/d/1gCEcEg62zb0QyOFt2xX4jWArLJYw3xoj/preview\" width=\"640\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat\u0026rsquo;s the article about creating midterm assessment questions in LaTeX. If you want to see other questions, please visit \u003ca href=\"https://www.aantriono.com/search/label/SOAL%20IX\" target=\"_blank\"\u003ehere\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Root Form Material Quiz", "date_published" : "2021-09-20T15:11:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kuis-materi-bentuk-akar/", "url" : "https://www.aantriono.com/en/kuis-materi-bentuk-akar/", "summary": " Introduction\nOn this occasion, we will share a quiz about the form of roots and powers of fractions. The material on roots and exponents of fractions is one of the materials taught in class IX middle school odd semester. You can view and study the material again here\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJ02Uz3m-FQICZdOThUtVPuV-dkOtu_ax0Xjs-aCx6chf4SbcaqRE1eTyV38sGpbuP2AZX2wDY-2PZVhnSLlPezLo4cmCRHBI6Kh4sHJ6WTQrHvJWfIZobr4zC8p5KfLVxaW3Ulb2gDTg/s1280/kuis+akar.jpeg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Kuis bentuk akar dan pangkat pecahan\" border=\"0\" data-original-height=\"1280\" data-original-width=\"1280\" height=\"400\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJ02Uz3m-FQICZdOThUtVPuV-dkOtu_ax0Xjs-aCx6chf4SbcaqRE1eTyV38sGpbuP2AZX2wDY-2PZVhnSLlPezLo4cmCRHBI6Kh4sHJ6WTQrHvJWfIZobr4zC8p5KfLVxaW3Ulb2gDTg/w400-h400/kuis+akar.jpeg\" title=\"Kuis bentuk akar\" width=\"400\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eOn this occasion, we will share a quiz about the form of roots and powers of fractions. The material on roots and exponents of fractions is one of the materials taught in class IX middle school odd semester. You can view and study the material again \u003ca href=\"https://www.aantriono.com/2021/07/bentuk-akar-dan-pangkat-pecahan.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e. The quiz written in this post is one way to measure understanding of the material.\u003c/p\u003e\n\u003cdiv class=\"postRelated\"\u003e\n \u003c!--[ Related title ]--\u003e\n\u003ch2 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also read\"\u003e\u003c/a\u003eAlso read\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2020/07/bilangan-berpangkat.html\" target=\"_blank\"\u003eRanked Numbers\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/07/bentuk-akar-dan-pangkat-pecahan.html\" target=\"_blank\"\u003eForm Roots and Exponents of Fractions\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n \u003c/ul\u003e\n\u003c/div\u003e\n\u003ch2 id=\"instruction\" class=\"headerLink\"\u003e\n \u003ca href=\"#instruction\" class=\"header-mark\" aria-label=\"Direct link to section: Instruction\"\u003e\u003c/a\u003eInstruction\u003c/h2\u003e\u003cp\u003eBefore you take this quiz, please read the following quiz instructions first:\u003c/p\u003e\n\u003col class=\"style1\"\u003e\n \u003cli\u003eClick the button Start to start the quiz.\u003c/li\u003e\n \u003cli\u003eWrite Name And Class you in the box provided.\u003c/li\u003e\n \u003cli\u003eClick the button Do itto go to the question menu.\u003c/li\u003e\n \u003cli\u003eClick on the button which is the correct answer.\u003c/li\u003e\n \u003cli\u003eClick on the sign \u003e to go to the next question, and click the sign \u003c to return to the previous question, or clickquestion menu to see the whole question.\u003c/li\u003e\n \u003cli\u003eThe quiz time is: 30 minutes.\u003c/li\u003e\n \u003cli\u003eThe KKM for this quiz is 70\u003c/li\u003e\n \u003cli\u003eClick the button Finished to end the quiz.\u003c/li\u003e\n \u003cli\u003eMark You will appear after clicking the finish button.\u003c/li\u003e\n \u003c/ol\u003e\n\u003ch3 id=\"quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Quiz\"\u003e\u003c/a\u003eQuiz\u003c/h3\u003e\u003cp\u003eBy : \u003ckbd\u003eAan Triono\u003c/kbd\u003e\u003c/p\u003e\n\u003c!--Tombol Mulai--\u003e\n\u003cdiv class=\"center\"\u003e\n \u003cbutton id=\"klik-mulai\"\u003eStart\u003c/button\u003e\n\u003c!--Box Input Nama dan Kelas--\u003e\n\u003cdiv id=\"input-data\"\u003e\n\u003c!--Box Soal--\u003e\n\u003cdiv id=\"bank-soal\"\u003e\n\u003cdiv class=\"waktu-wrap\"\u003e\n\u003cdiv class=\"data-peserta\"\u003e\n\u003cdiv class=\"nama-peserta\"\u003e\n\u003cdiv class=\"kelas-peserta\"\u003e\n\u003c!--Waktu Soal, Ganti berdasarkan jam--\u003e\n\u003cdiv data-waktu=\"0.5\" id=\"waktu-soal\"\u003e\n \u003c!--Jumlah Soal, Ganti berdasrkan jumlah soal--\u003e\n\u003cdiv class=\"bank-soal-wrap\" data-soal=\"10\"\u003e\n\u003c!--Pertanyaan dan Jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e1. The simple form of the form $\\sqrt{9p^5q}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3p\\sqrt{pq}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$3p^2\\sqrt{pq}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3p^3\\sqrt{pq}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3p^4\\sqrt{pq}$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Penutup kode pertanyaan dan jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e2. The simple form of the form $\\sqrt{8}+\\sqrt{32}+\\sqrt{72}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$12\\sqrt{2}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$12\\sqrt{3}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$12\\sqrt{5}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$12\\sqrt{7}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e3. The simple form of $\\sqrt{p}+\\sqrt{4p}-\\sqrt{9p}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$0$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Tombol Soal, Sesuaikan dengan jumlah soal--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e4. The simple form of $\\dfrac{\\sqrt{6}\\times \\sqrt{2}}{\\sqrt{3}}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$0$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e5. The simple form of $\\dfrac{\\sqrt{24}+\\sqrt{54}-\\sqrt{150}}{\\sqrt{96}}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$0$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e6. The simple form of $(\\sqrt{3}-\\sqrt{2})(\\sqrt{3}-5\\sqrt{2})$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$13-5\\sqrt{6}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$13-6\\sqrt{6}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$13-7\\sqrt{6}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$13-8\\sqrt{6}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e7. The rational form of $\\dfrac{3}{\\sqrt{2}}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$\\dfrac{3\\sqrt{2}}{2}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3\\sqrt{2}}{3}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3\\sqrt{3}}{2}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3\\sqrt{2}}{3}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e8. The rational form of $\\sqrt{0,1}$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{4}{10}\\sqrt{10}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{3}{10}\\sqrt{10}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{2}{10}\\sqrt{10}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$\\dfrac{1}{10}\\sqrt{10}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e9. The value of $x$ that satisfies $9^{2x-3}=27$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{9}{6}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{9}{5}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$\\dfrac{9}{4}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\dfrac{9}{2}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e10. The value of $x$ that satisfies $(\\sqrt[3]{2})^{x+10}=8$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$0$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$-1$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv id=\"show-hide-box-soal\"\u003e\n\u003cdiv class=\"tombol-show-hide lihat-soal\"\u003e\n closed\n\u003cdiv class=\"menu-tombol-soal\"\u003e\n\u003cdiv class=\"tombol-soal\" data-nomor=\"1\"\u003e1\n\u003cdiv class=\"tombol-soal\" data-nomor=\"2\"\u003e2\n\u003cdiv class=\"tombol-soal\" data-nomor=\"3\"\u003e3\n\u003cdiv class=\"tombol-soal\" data-nomor=\"4\"\u003e4\n\u003cdiv class=\"tombol-soal\" data-nomor=\"5\"\u003e5\n\u003cdiv class=\"tombol-soal\" data-nomor=\"6\"\u003e6\n\u003cdiv class=\"tombol-soal\" data-nomor=\"7\"\u003e7\n\u003cdiv class=\"tombol-soal\" data-nomor=\"8\"\u003e8\n\u003cdiv class=\"tombol-soal\" data-nomor=\"9\"\u003e9\n\u003cdiv class=\"tombol-soal\" data-nomor=\"10\"\u003e10\n\u003cdiv id=\"menu-soal\"\u003e\n\u003cdiv class=\"next-prev-soal\"\u003e\n\u003cdiv class=\"prev\" id=\"prev-soal\"\u003e\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M15.41,16.58L10.83,12L15.41,7.41L14,6L8,12L14,18L15.41,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n Previously\n\u003cdiv class=\"next\" id=\"next-soal\"\u003e\n Furthermore\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M8.59,16.58L13.17,12L8.59,7.41L10,6L16,12L10,18L8.59,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n\u003cdiv class=\"menu-pilihan-soal\"\u003e\n \u003cbutton class=\"lihat-soal\" id=\"lihat-soal\"\u003eQuestion Menu\u003c/button\u003e\n\u003cdiv class=\"tombol-soal-selesai\"\u003e\n \u003cbutton id=\"soal-selesai\"\u003eFinished\u003c/button\u003e\n\u003cdiv id=\"lihat-hasil-nilai\"\u003e\n\u003cdiv id=\"hasil-pengerjaan\"\u003e\n\u003cp class=\"info-keterangan-hasil\"\u003e*Please screenshot the results and send via WhatsApp **\u003ca href=\"https://www.aantriono.com/p/contact.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\n\u003cp\u003e**, or you can also share it in the comments, and if you remedial you can repeat it again by clicking **\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-bentuk-akar.html\"\u003erepeat\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e**\u003c/p\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n", "language": "en" }, { "title" : "Quiz on Numbers for Class IX SMP/MTs", "date_published" : "2021-09-18T04:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kuis-materi-bilangan-berpangkat/", "url" : "https://www.aantriono.com/en/kuis-materi-bilangan-berpangkat/", "summary": " Introduction\nPower numbers are one of the materials studied in class IX odd semester. To learn it, please reopen the material rank number\n. The basics of exponent number material have previously been studied in class VII in whole number material. To measure understanding regarding the material on exponents, please take the following quiz on exponents.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXQzjxPgb_vFmCbCKF9ZNjwP6ldV01862QsFTCLTgrfFs4TVRhbUbhZjzRnfnapHixJVAPFwMJbckLaj6DHXIeTBPwGbJg9UUwlOpInlhyHEIaIqBYf-UEt_KxLKTb5Wnp_TIw5UArH00/s1280/photo1631937271.jpeg\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Kuis materi bilangan berpangkat kelas 9\" border=\"0\" data-original-height=\"1280\" data-original-width=\"1280\" height=\"400\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXQzjxPgb_vFmCbCKF9ZNjwP6ldV01862QsFTCLTgrfFs4TVRhbUbhZjzRnfnapHixJVAPFwMJbckLaj6DHXIeTBPwGbJg9UUwlOpInlhyHEIaIqBYf-UEt_KxLKTb5Wnp_TIw5UArH00/w400-h400/photo1631937271.jpeg\" title=\"Kuis materi bilangan berpangkat\" width=\"400\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003ePower numbers are one of the materials studied in class IX odd semester. To learn it, please reopen the material \u003ca href=\"https://www.aantriono.com/2020/07/bilangan-berpangkat.html\" target=\"_blank\"\u003erank number\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e. The basics of exponent number material have previously been studied in class VII in whole number material. To measure understanding regarding the material on exponents, please take the following quiz on exponents.\u003c/p\u003e\n\u003ch2 id=\"instruction\" class=\"headerLink\"\u003e\n \u003ca href=\"#instruction\" class=\"header-mark\" aria-label=\"Direct link to section: Instruction\"\u003e\u003c/a\u003eInstruction\u003c/h2\u003e\u003col class=\"style1\"\u003e\n \u003cli\u003eClick the button Start to start the quiz.\u003c/li\u003e\n \u003cli\u003eWrite Name And Classyou.\u003c/li\u003e\n \u003cli\u003eClick the button Continueto go to the questions and answers.\u003c/li\u003e\n \u003cli\u003eClick on the button which is the correct answer.\u003c/li\u003e\n \u003cli\u003eThe quiz time is: 30 minutes (1800 seconds).\u003c/li\u003e\n \u003cli\u003eThe KKM for this quiz is 70\u003c/li\u003e\n \u003cli\u003eClick the button Finished to end the quiz.\u003c/li\u003e\n \u003cli\u003eMark You will appear after clicking the finish button.\u003c/li\u003e\n \u003cli\u003eClick on the text*Try again* to repeat the quiz.\u003c/li\u003e\n \u003c/ol\u003e\n\u003ch2 id=\"quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Quiz\"\u003e\u003c/a\u003eQuiz\u003c/h2\u003e\u003cp\u003eBy : \u003ckbd\u003eAan Triono\u003c/kbd\u003e\u003c/p\u003e\n\u003c!--[tombol mulai]--\u003e\n\u003cdiv style='text-align:center;'\u003e\n\u003cbutton id='wc-qws-start'\u003eStart\u003c/button\u003e\n\u003c!--[tombol mulai and]--\u003e\n\u003c!--[form input nama mahasiswa dan kelas]--\u003e\n\u003cdiv class='wc-qws-mhs hidden'\u003e\n\u003cinput id='wc-qws-nmhs' type='text' placeholder='Nama siswa' autocomplete='off'/\u003e\n\u003cinput id='wc-qws-kls' type='text' placeholder='Kelas' autocomplete='off'/\u003e\n\u003cbutton id='wc-qws-lnj'\u003eContinue\u003c/button\u003e\n\u003c!--[form input nama mahasiswa dan kelas and]--\u003e\n\u003c!--[kuis waktu ubah anggka 1800 dengan waktu yang di inginkan]--\u003e\n\u003cdiv class=\"wc-qws-wkt hidden\"\u003eRemaining time: 1800 second\n\u003c!--[kuis konten]--\u003e\n\u003c!--[sesuaikan anggka 5 pada data-quis dengan jumlah kuis yang ada]--\u003e\n\u003cdiv class=\"wc-qws-knt hidden\" data-quis=\"10\"\u003e\n\u003c!--[kuis 1]--\u003e\n\u003cform\u003e\n\u003cp\u003e**1**The result of $49^\\frac{3}{2}$ is...\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e49\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e343\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e2,401\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e16,807\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 1 and]--\u003e\n\u003c!--[kuis 2]--\u003e\n\u003cform\u003e\n\u003cp\u003e**2** The value of $8^\\frac{4}{3}+9^\\frac{3}{2}$ is...\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e23\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e33\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e43\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e53\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 2 and]--\u003e\n\u003c!--[kuis 3]--\u003e\n\u003cform\u003e\n\u003cp\u003e**3** If $5.23^2=27.3529$ then $52.3^2=\\cdots$\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e273,529\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e2,735.29\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e27,352.9\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e273,529\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 3 and]--\u003e\n\u003c!--[kuis 4]--\u003e\n\u003cform\u003e\n\u003cp\u003e**4** The result of $1,000^0+100^0+10^0+1^0+(\\dfrac{1}{10})^0=\\cdots$\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e1,112\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e1,111.1\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e5\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e1\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 4 and]--\u003e\n\u003c!--[kuis 5]--\u003e\n\u003cform\u003e\n\u003cp\u003e**5** The result is $(1.25\\times 10^3)\\times (8\\times 10^{-2})=\\cdots$\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e1\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e10\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e100\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e1,000\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 5 and]--\u003e\n\u003c!--[tambahkan kuis selanjutnya di sini dan jangan lupa untuk mengubah data-quis di atas]--\u003e\n\u003c!--[kuis 6]--\u003e\n\u003cform\u003e\n\u003cp\u003e**6** The result of $(-8m^2 n^3 ) \\times (2k^3 n^4 )=\\cdots$\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e$-16k^3 m^2 n^{12}$\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e $-16k^3 m^2 n^7$\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e$16k^3 m^2 n^7$ \u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e$16k^3 m^2 n^7$\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 6 and]--\u003e\n\u003c!--[kuis 7]--\u003e\n\u003cform\u003e\n\u003cp\u003e**7** The result of $\\dfrac{4^\\frac{-2}{3}\\times 4^\\frac{-4}{3}}{4^{-3}}=\\cdots$\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e1\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e2\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e3\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e4\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 7 and]--\u003e\n\u003c!--[kuis 8]--\u003e\n\u003cform\u003e\n\u003cp\u003e**8**If $\\dfrac{64}{729}=(\\dfrac{2}{3})^{x-2}$ then the value of $x^2+1=\\cdots$\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e65\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e55\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e45\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e35\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 8 and]--\u003e\n \u003c!--[kuis 9]--\u003e\n\u003cform\u003e\n\u003cp\u003e**9**Given $(\\dfrac{3}{4})^9=(\\dfrac{4}{3})^a$ then the value of $a$ is...\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e$-5$\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e$-7$\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e$-9$\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e$-11$\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 9 and]--\u003e\n\u003c!--[kuis 10]--\u003e\n\u003cform\u003e\n\u003cp\u003e**10**If $(\\dfrac{3}{4})^{x-2}=(\\dfrac{81}{256})$ then the value of $x=\\cdots$\u003c/p\u003e\n\u003c!--[value=\"true\" untuk jawaban benar dan value=\"false\" untuk jawaban salah ]--\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e4\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"true\" /\u003e6\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e8\u003c/label\u003e\n\u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"false\" /\u003e10\u003c/label\u003e\n\u003c/form\u003e\n\u003c!--[kuis 10 and]--\u003e\n\u003c!--[tombol selesai]--\u003e\n\u003cdiv style='text-align:center;'\u003e\n\u003cbutton id='wc-qws-fns'\u003eFinished\u003c/button\u003e\n\u003c!--[tombol selesai and]--\u003e\n\u003c!--[kuis konten and]--\u003e\n\u003c!--[tampilan selesai]--\u003e\n\u003cdiv class='wc-qws-sls hidden'\u003e\n\u003ch3 id=\"congratulations-you-have-completed-this-quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#congratulations-you-have-completed-this-quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Congratulations, you have completed this quiz.\"\u003e\u003c/a\u003eCongratulations, you have completed this quiz.\u003c/h3\u003e\u003cp\u003eStudent name: \u003c/p\u003e\n\u003cp\u003eClass: \u003c/p\u003e\n\u003cp\u003eCorrect Answer: \u003c/p\u003e\n\u003cp\u003eWrong Answer: \u003c/p\u003e\n\u003cp\u003eNot Answered: \u003c/p\u003e\n\u003cp\u003eMark: \u003c/p\u003e\n\u003cp\u003eAfter you have finished taking the CSS material quiz, how did you get your score? successful or need to repeat? \u003ca id='wc-qws-rst' href='#'\u003etry again\u003c/a\u003e\n\u003c/p\u003e\n\u003c!--[tampilan selesai and]--\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n", "language": "en" }, { "title" : "Quadratic Equations Material Quiz", "date_published" : "2021-09-14T03:37:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kuis-materi-persamaan-kuadrat/", "url" : "https://www.aantriono.com/en/kuis-materi-persamaan-kuadrat/", "summary": " IntroductionThe previous posts discussed:\nQuadratic Equations Discriminant Function Constructing Quadratic Equations The following quiz is provided to help readers measure their understanding of those topics.\nInstruction Click Start to begin the quiz. The quiz contains 10 multiple-choice questions. Choose the correct answer for each question. The time limit is 30 minutes or 1800 seconds. The minimum passing score is 70. Click Finished to end the quiz. Your score will appear after the quiz is completed. QuizBy : Aan Triono\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThe previous posts discussed:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/08/persamaan-kuadrat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eQuadratic Equations\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/08/fungsi-diskriminan/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eDiscriminant Function\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/09/menyusun-persamaan-kuadrat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eConstructing Quadratic Equations\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThe following quiz is provided to help readers measure their understanding of those topics.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfZNI4-QTagPajz8y-WhdV3RMOW5YkZgHi_2TaZnSW3sEka9u2wiClVhgQ2SFRmjfWzh_l4ELCUGnuVjxhPLmtfI8b6b5hy8JKUzNn4b4pTOPqrX_qPfmYcYAOE_AajPdgY1bNQrti5dE/w640-h365/kuis+pk.jpeg' alt=\"Quadratic equation quiz\" \u003e\u003c/p\u003e\n\u003ch2 id=\"instruction\" class=\"headerLink\"\u003e\n \u003ca href=\"#instruction\" class=\"header-mark\" aria-label=\"Direct link to section: Instruction\"\u003e\u003c/a\u003eInstruction\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eClick \u003ccode\u003eStart\u003c/code\u003e to begin the quiz.\u003c/li\u003e\n\u003cli\u003eThe quiz contains 10 multiple-choice questions.\u003c/li\u003e\n\u003cli\u003eChoose the correct answer for each question.\u003c/li\u003e\n\u003cli\u003eThe time limit is 30 minutes or 1800 seconds.\u003c/li\u003e\n\u003cli\u003eThe minimum passing score is 70.\u003c/li\u003e\n\u003cli\u003eClick \u003ccode\u003eFinished\u003c/code\u003e to end the quiz.\u003c/li\u003e\n\u003cli\u003eYour score will appear after the quiz is completed.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Quiz\"\u003e\u003c/a\u003eQuiz\u003c/h3\u003e\u003cp\u003eBy : \u003ckbd\u003eAan Triono\u003c/kbd\u003e\u003c!--Tombol Mulai--\u003e\u003c/p\u003e\n\u003cdiv class=\"center\"\u003e\n \u003cbutton id=\"klik-mulai\"\u003eStart\u003c/button\u003e\n\u003c!--Box Input Nama dan Kelas--\u003e\n\u003cdiv id=\"input-data\"\u003e\n\u003c!--Box Soal--\u003e\n\u003cdiv id=\"bank-soal\"\u003e\n\u003cdiv class=\"waktu-wrap\"\u003e\n\u003cdiv class=\"data-peserta\"\u003e\n\u003cdiv class=\"nama-peserta\"\u003e\n\u003cdiv class=\"kelas-peserta\"\u003e\n\u003c!--Waktu Soal, Ganti berdasarkan jam--\u003e\n\u003cdiv data-waktu=\"0.5\" id=\"waktu-soal\"\u003e\n \u003c!--Jumlah Soal, Ganti berdasrkan jumlah soal--\u003e\n\u003cdiv class=\"bank-soal-wrap\" data-soal=\"10\"\u003e\n\u003c!--Pertanyaan dan Jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e1. The equation $x^2-1=2(x-3)$ has the general form $ax^2+bx+c=0$. Value $a+b+c=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$4$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Penutup kode pertanyaan dan jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e2. The coefficient $x$ of the equation $x+\\dfrac{3}{x-1}=2$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$-3$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e3. If $x=2$ is one of the roots of the equation $x^2+5x+c=0$, then the value $c=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$14$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$6$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-6$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$-14$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Tombol Soal, Sesuaikan dengan jumlah soal--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e4. If $x=3$ is one of the roots of the equation $x^2+2x+c=0$, then the other root is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-15$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$-5$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$1$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e5. The set of solutions to the equation $x^2+7x+12=0$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\{3, 4\\}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\{3, -4\\}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$\\{-3, -4\\}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\{6-, 2\\}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e6. The types of roots of the equation $x^2-6x+9=0$ are...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003ereal is different\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003ereal different signs\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003ereal twins\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003eimaginary\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e7. So that the equation $x^2+px+25=0$ has twin real roots, then the value $p=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$\\pm 15$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$\\pm 10$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$5$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-5$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e8. The roots of the equation $x^2-12=4x$ are $x_1$ and $x_2$. The value of $x_1+x_2=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$6$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$4$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$-4$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e9. A quadratic equation whose roots are twice the roots of the equation $x^2-3x-28=0$ is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$x^2-6x-112=0$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$x^2-6x-56=0$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2x^2-6x-56=0$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2x^2-3x-56=0$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e10. A rectangle is 3 cm longer than its width. If the area is $180\\ \\text{cm}^2$, then the perimeter of the rectangle is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$56\\ \\text{cm}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$54\\ \\text{cm}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$48\\ \\text{cm}$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$42\\ \\text{cm}$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv id=\"show-hide-box-soal\"\u003e\n\u003cdiv class=\"tombol-show-hide lihat-soal\"\u003e\n closed\n\u003cdiv class=\"menu-tombol-soal\"\u003e\n\u003cdiv class=\"tombol-soal\" data-nomor=\"1\"\u003e1\n\u003cdiv class=\"tombol-soal\" data-nomor=\"2\"\u003e2\n\u003cdiv class=\"tombol-soal\" data-nomor=\"3\"\u003e3\n\u003cdiv class=\"tombol-soal\" data-nomor=\"4\"\u003e4\n\u003cdiv class=\"tombol-soal\" data-nomor=\"5\"\u003e5\n\u003cdiv class=\"tombol-soal\" data-nomor=\"6\"\u003e6\n\u003cdiv class=\"tombol-soal\" data-nomor=\"7\"\u003e7\n\u003cdiv class=\"tombol-soal\" data-nomor=\"8\"\u003e8\n\u003cdiv class=\"tombol-soal\" data-nomor=\"9\"\u003e9\n\u003cdiv class=\"tombol-soal\" data-nomor=\"10\"\u003e10\n\u003cdiv id=\"menu-soal\"\u003e\n\u003cdiv class=\"next-prev-soal\"\u003e\n\u003cdiv class=\"prev\" id=\"prev-soal\"\u003e\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M15.41,16.58L10.83,12L15.41,7.41L14,6L8,12L14,18L15.41,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n Previously\n\u003cdiv class=\"next\" id=\"next-soal\"\u003e\n Furthermore\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M8.59,16.58L13.17,12L8.59,7.41L10,6L16,12L10,18L8.59,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n\u003cdiv class=\"menu-pilihan-soal\"\u003e\n \u003cbutton class=\"lihat-soal\" id=\"lihat-soal\"\u003eQuestion Menu\u003c/button\u003e\n\u003cdiv class=\"tombol-soal-selesai\"\u003e\n \u003cbutton id=\"soal-selesai\"\u003eFinished\u003c/button\u003e\n\u003cdiv id=\"lihat-hasil-nilai\"\u003e\n\u003cdiv id=\"hasil-pengerjaan\"\u003e\n\u003cp class=\"info-keterangan-hasil\"\u003e*Please screenshot the results and send via WhatsApp **\u003ca href=\"https://www.aantriono.com/p/contact.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\n\u003cp\u003e**, or you can also share it in the comments, and if you remedial you can repeat it again by clicking **\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-persamaan-kuadrat.html\"\u003erepeat\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e**\u003c/p\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n", "language": "en" }, { "title" : "Composing New Quadratic Equations", "date_published" : "2021-09-13T08:10:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menyusun-persamaan-kuadrat/", "url" : "https://www.aantriono.com/en/menyusun-persamaan-kuadrat/", "summary": " IntroductionIn previous lessons on quadratic equations and the discriminant, we learned how to find roots by factoring, completing the square, and using the quadratic formula. The roots of a quadratic equation may be different, equal, real, or imaginary.\nConsider this problem. Suppose there are two consecutive even numbers whose product is 224. If the first number is xxx, then the second is x+2x+2x+2. The mathematical model is:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn previous lessons on quadratic equations and the discriminant, we learned how to find roots by factoring, completing the square, and using the quadratic formula. The roots of a quadratic equation may be different, equal, real, or imaginary.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioKtre9RrxdsXcS547YHz0JPT703sjdx3ggnEKrfLHZY8a_GSZEPioOf6MpSjGaDGFqIt2RQP1H1ZjWGqdtgnIs6A7qBbYgOydhvC2gMEFWzG4hdkDaolwhp9XsV0kuojBZ4otvO7cuvg/w640-h640/persamaan+kuadrat.jpeg' alt=\"Composing quadratic equations\" \u003e\u003c/p\u003e\n\u003cp\u003eConsider this problem. Suppose there are two consecutive even numbers whose product is 224. If the first number is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then the second is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex+2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. The mathematical model is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e224\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx(x+2)=224\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e224\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e224\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2+2x-224=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e224\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSolving the equation gives \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, so the two numbers are \u003cstrong\u003e14 and 16\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eNow, what if the roots of a quadratic equation are already known? How do we form the equation?\u003c/p\u003e\n\u003ch2 id=\"constructing-a-quadratic-equation-from-its-roots\" class=\"headerLink\"\u003e\n \u003ca href=\"#constructing-a-quadratic-equation-from-its-roots\" class=\"header-mark\" aria-label=\"Direct link to section: Constructing a Quadratic Equation from Its Roots\"\u003e\u003c/a\u003eConstructing a Quadratic Equation from Its Roots\u003c/h2\u003e\u003cp\u003eIf a quadratic equation has roots \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then it can be formed by:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(x-x_1)(x-x_2)=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eDetermine the quadratic equation whose roots are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"answer\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h3\u003e\u003ch4 id=\"a-roots--and\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-roots--and\" class=\"header-mark\" aria-label=\"Direct link to section: a. Roots −2-2−2 and 666\"\u003e\u003c/a\u003ea. Roots \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h4\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n(x-x_1)(x-x_2)\u0026amp;=0\\\\\n(x-(-2))(x-6)\u0026amp;=0\\\\\n(x+2)(x-6)\u0026amp;=0\\\\\nx^2-6x+2x-12\u0026amp;=0\\\\\nx^2-4x-12\u0026amp;=0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.5482em;vertical-align:-3.5241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0241em;\"\u003e\u003cspan style=\"top:-6.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e))\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0241em;\"\u003e\u003cspan style=\"top:-6.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the required quadratic equation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2-4x-12=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"b-roots--and\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-roots--and\" class=\"header-mark\" aria-label=\"Direct link to section: b. Roots 13\\dfrac{1}{3}31​ and 12\\dfrac{1}{2}21​\"\u003e\u003c/a\u003eb. Roots \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h4\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n(x-x_1)(x-x_2)\u0026amp;=0\\\\\n\\left(x-\\frac{1}{3}\\right)\\left(x-\\frac{1}{2}\\right)\u0026amp;=0\\\\\nx^2-\\frac{1}{3}x-\\frac{1}{2}x+\\frac{1}{6}\u0026amp;=0\\\\\nx^2-\\frac{5}{6}x+\\frac{1}{6}\u0026amp;=0\\\\\n6x^2-5x+1\u0026amp;=0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:10.339em;vertical-align:-4.9195em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.4195em;\"\u003e\u003cspan style=\"top:-8.0295em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.9195em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.348em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.0406em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.8095em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.9195em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.4195em;\"\u003e\u003cspan style=\"top:-8.0295em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.9195em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.348em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.0406em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.8095em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.9195em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the required quadratic equation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n6x^2-5x+1=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"related-reading\" class=\"headerLink\"\u003e\n \u003ca href=\"#related-reading\" class=\"header-mark\" aria-label=\"Direct link to section: Related Reading\"\u003e\u003c/a\u003eRelated Reading\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/08/persamaan-kuadrat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eQuadratic Equations\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/08/fungsi-diskriminan/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eDiscriminant Function\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"constructing-a-quadratic-equation-from-the-sum-and-product-of-its-roots\" class=\"headerLink\"\u003e\n \u003ca href=\"#constructing-a-quadratic-equation-from-the-sum-and-product-of-its-roots\" class=\"header-mark\" aria-label=\"Direct link to section: Constructing a Quadratic Equation from the Sum and Product of Its Roots\"\u003e\u003c/a\u003eConstructing a Quadratic Equation from the Sum and Product of Its Roots\u003c/h2\u003e\u003cp\u003eIf the roots are \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n(x-x_1)(x-x_2)\u0026amp;=0\\\\\nx^2-x_1x-x_2x+x_1x_2\u0026amp;=0\\\\\nx^2-(x_1+x_2)x+x_1x_2\u0026amp;=0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, if the sum \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex_1+x_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the product \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex_1x_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are known, the equation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2-(x_1+x_2)x+x_1x_2=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFrom the quadratic formula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx_1=\\frac{-b+\\sqrt{b^2-4ac}}{2a}\n\\quad \\text{and} \\quad\nx_2=\\frac{-b-\\sqrt{b^2-4ac}}{2a}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2764em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5904em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9134em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan 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style=\"top:-2.8734em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.9755em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4755em;\"\u003e\u003cspan style=\"top:-4.4755em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3819em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.9755em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThus, for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eax^2+bx+c=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx_1+x_2=\\frac{-b}{a}\n\\quad \\text{and} \\quad\nx_1x_2=\\frac{c}{a}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx_1+x_2=7 \\quad \\text{and} \\quad x_1x_2=10\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eDetermine the quadratic equation.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx^2-(x_1+x_2)x+x_1x_2\u0026amp;=0\\\\\nx^2-7x+10\u0026amp;=0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0482em;vertical-align:-1.2741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\beta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the roots of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2x^2-4x-12=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003efind:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha+\\beta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\\beta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\\beta^2+\\alpha^2\\beta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha^2+\\beta^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"answer-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-1\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\alpha+\\beta\u0026amp;=\\frac{-b}{a}\n=\\frac{-(-4)}{2}\n=2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.413em;vertical-align:-0.9565em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4565em;\"\u003e\u003cspan style=\"top:-3.4565em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.427em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9565em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4565em;\"\u003e\u003cspan style=\"top:-3.4565em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.427em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.427em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9565em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\alpha\\beta\u0026amp;=\\frac{c}{a}\n=\\frac{-12}{2}\n=-6\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.3074em;vertical-align:-0.9037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4037em;\"\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4037em;\"\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\alpha\\beta^2+\\alpha^2\\beta\n\u0026amp;=\\alpha\\beta(\\alpha+\\beta)\\\\\n\u0026amp;=(-6)(2)\\\\\n\u0026amp;=-12\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5241em;vertical-align:-2.0121em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\alpha^2+\\beta^2\n\u0026amp;=(\\alpha+\\beta)^2-2\\alpha\\beta\\\\\n\u0026amp;=2^2-2(-6)\\\\\n\u0026amp;=16\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example 3\"\u003e\u003c/a\u003eExample 3\u003c/h3\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\beta\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the roots of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2x^2-14x+12=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003edetermine a new quadratic equation whose roots are \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha+2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\beta+2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"answer-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-2\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h3\u003e\u003cp\u003eFrom \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2-14x+12=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, we have:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na=2,\\quad b=-14,\\quad c=12\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\alpha+\\beta=\\frac{-b}{a}=\\frac{14}{2}=7\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\alpha\\beta=\\frac{c}{a}=\\frac{12}{2}=6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe new sum is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(\\alpha+2)+(\\beta+2)=7+4=11\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe new product is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(\\alpha+2)(\\beta+2)=6+2(7)+4=24\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the new quadratic equation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eβ\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx^2-[(\\alpha+2)+(\\beta+2)]x+(\\alpha+2)(\\beta+2)\u0026amp;=0\\\\\nx^2-11x+24\u0026amp;=0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0482em;vertical-align:-1.2741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e[(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)]\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05278em;\"\u003eβ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"solving-problems-related-to-quadratic-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#solving-problems-related-to-quadratic-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Solving Problems Related to Quadratic Equations\"\u003e\u003c/a\u003eSolving Problems Related to Quadratic Equations\u003c/h2\u003e\u003cp\u003eMany daily-life problems can be modeled using quadratic equations. The general steps are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003elet one unknown be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003eform an equation in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003esolve the equation,\u003c/li\u003e\n\u003cli\u003echoose the solution that fits the context.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"example-1-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eThe difference between two whole numbers is 2 and their product is 168. Determine the two numbers.\u003c/p\u003e\n\u003ch3 id=\"answer-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-3\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h3\u003e\u003cp\u003eLet the first number be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then the second is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex-2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e168\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e168\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" 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width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx(x-2)\u0026amp;=168\\\\\nx^2-2x\u0026amp;=168\\\\\nx^2-2x-168\u0026amp;=0\\\\\n(x+12)(x-14)\u0026amp;=0\\\\\nx\u0026amp;=-12 \\quad \\text{or} \\quad x=14\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.5482em;vertical-align:-3.5241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0241em;\"\u003e\u003cspan style=\"top:-6.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e168\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0241em;\"\u003e\u003cspan style=\"top:-6.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e168\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e168\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is not a whole number, the valid solution is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eTherefore, the two numbers are \u003cstrong\u003e12 and 14\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-2-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eMr. Harun\u0026rsquo;s garden is rectangular. Its length is 3 meters more than its width. If the area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e108\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e m\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e108 \\text{ m}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e108\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e m\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, determine its perimeter.\u003c/p\u003e\n\u003ch3 id=\"answer-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-4\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h3\u003e\u003cp\u003eLet the length be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then the width is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep-3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e108\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e108\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e108\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n108\u0026amp;=p(p-3)\\\\\n108\u0026amp;=p^2-3p\\\\\np^2-3p-108\u0026amp;=0\\\\\n(p-12)(p+9)\u0026amp;=0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0482em;vertical-align:-2.7741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.4341em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e108\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e108\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e108\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.4341em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\np=12 \\quad \\text{or} \\quad p=-9\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eOnly \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep=12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is valid. Then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003el\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nl=12-3=9\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.01968em;\"\u003el\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe perimeter is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003el\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e42\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nK\u0026amp;=2(p+l)\\\\\n\u0026amp;=2(12+9)\\\\\n\u0026amp;=42\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.01968em;\"\u003el\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e42\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the perimeter of Mr. Harun\u0026rsquo;s garden is \u003cstrong\u003e42 meters\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eAfter studying quadratic equations, the discriminant function, and composing quadratic equations, continue to the evaluation below:\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"/2021/09/kuis-materi-persamaan-kuadrat/\" rel=\"\"\u003eQuadratic Equation Quiz\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat concludes this material on composing quadratic equations for Grade IX SMP/MTs.\u003c/p\u003e\n", "language": "en" }, { "title" : "Set Material Quiz", "date_published" : "2021-09-12T15:09:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kuis-materi-himpunan/", "url" : "https://www.aantriono.com/en/kuis-materi-himpunan/", "summary": " IntroductionIn the previous post, I wrote about Definition of Set\n, Presenting Relationships Between Sets Using Venn Diagrams\n, And Presenting Set Operations in a Venn Diagram\n. Now, I will give a quiz related to the set material. This is intended so that readers can measure their understanding in studying material about the set which is divided into these three posts.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn the previous post, I wrote about \u003ca href=\"https://www.aantriono.com/2021/08/pengertian-himpunan.html\" target=\"_blank\"\u003eDefinition of Set\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e, \u003ca href=\"https://www.aantriono.com/2021/08/menyajikan-hubungan-antarhimpunan-dengan-diagram-venn.html\" target=\"_blank\"\u003ePresenting Relationships Between Sets Using Venn Diagrams\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e, And \u003ca href=\"https://www.aantriono.com/2021/09/menyajikan-operasi-himpunan-dalam-diagram-venn.html\" target=\"_blank\"\u003ePresenting Set Operations in a Venn Diagram\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e. Now, I will give a quiz related to the set material. This is intended so that readers can measure their understanding in studying material about the set which is divided into these three posts.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsjUNHHeaNwAzCoMbAVJtLzjXqwQcjd-2YKlUV_HcLG0SeVXjnKxNxWx-7BzDlOI9nV3-eFFJ6Z3hlT7pYRE8jTeUer4PbsiKlPdZAc3iptYxJPZbGd38B-dEjQ4-6WFArwpPOV8qPIlE/s1280/kuis+himpunan.jpeg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"kuis irisan himpunan\" border=\"0\" data-original-height=\"1280\" data-original-width=\"1280\" height=\"400\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsjUNHHeaNwAzCoMbAVJtLzjXqwQcjd-2YKlUV_HcLG0SeVXjnKxNxWx-7BzDlOI9nV3-eFFJ6Z3hlT7pYRE8jTeUer4PbsiKlPdZAc3iptYxJPZbGd38B-dEjQ4-6WFArwpPOV8qPIlE/w400-h400/kuis+himpunan.jpeg\" title=\"irisan himpunan\" width=\"400\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"instructions-for-taking-the-quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#instructions-for-taking-the-quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Instructions for Taking the Quiz\"\u003e\u003c/a\u003eInstructions for Taking the Quiz\u003c/h2\u003e\u003col class=\"style1\"\u003e\n \u003cli\u003eClick the button Start to start the quiz.\u003c/li\u003e\n \u003cli\u003eWrite Name And Class you in the box provided.\u003c/li\u003e\n \u003cli\u003eClick the button Do itto go to the question menu.\u003c/li\u003e\n \u003cli\u003eClick on the button which is the correct answer.\u003c/li\u003e\n \u003cli\u003eClick on the sign \u003e to go to the next question, and click the sign \u003c to return to the previous question, or clickquestion menu to see the whole question.\u003c/li\u003e\n \u003cli\u003eThe quiz time is: 30 minutes.\u003c/li\u003e\n \u003cli\u003eThe KKM for this quiz is 70\u003c/li\u003e\n \u003cli\u003eClick the button Finished to end the quiz.\u003c/li\u003e\n \u003cli\u003eMark You will appear after clicking the finish button.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"postRelated\"\u003e\n \u003c!--[ Related title ]--\u003e\n\u003ch2 id=\"also-read\" class=\"headerLink\"\u003e\n \u003ca href=\"#also-read\" class=\"header-mark\" aria-label=\"Direct link to section: Also read\"\u003e\u003c/a\u003eAlso read\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-bentuk-akar.html\" target=\"_blank\"\u003eRoot Form Material Quiz\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-bilangan-berpangkat.html\" target=\"_blank\"\u003eRanked Numbers Material Quiz\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#\"\u003e\u003c/a\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-persamaan-kuadrat.html\" target=\"_blank\"\u003ePower Equation Material Quiz\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n \u003c/ul\u003e\n\u003c/div\u003e\n\u003ch3 id=\"quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Quiz\"\u003e\u003c/a\u003eQuiz\u003c/h3\u003e\u003cp\u003eBy : \u003ckbd\u003eAan Triono\u003c/kbd\u003e\u003c/p\u003e\n\u003c!--Tombol Mulai--\u003e\n\u003cdiv class=\"center\"\u003e\n \u003cbutton id=\"klik-mulai\"\u003eStart\u003c/button\u003e\n\u003c!--Box Input Nama dan Kelas--\u003e\n\u003cdiv id=\"input-data\"\u003e\n\u003c!--Box Soal--\u003e\n\u003cdiv id=\"bank-soal\"\u003e\n\u003cdiv class=\"waktu-wrap\"\u003e\n\u003cdiv class=\"data-peserta\"\u003e\n\u003cdiv class=\"nama-peserta\"\u003e\n\u003cdiv class=\"kelas-peserta\"\u003e\n\u003c!--Waktu Soal, Ganti berdasarkan jam--\u003e\n\u003cdiv data-waktu=\"0.5\" id=\"waktu-soal\"\u003e\n \u003c!--Jumlah Soal, Ganti berdasrkan jumlah soal--\u003e\n\u003cdiv class=\"bank-soal-wrap\" data-soal=\"10\"\u003e\n\u003c!--Pertanyaan dan Jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e1. It is known that $P=\\{\\text{factor of 60 is divisible by 3}\\}$. The correct statement below is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$6\\not \\in P$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$9\\in P$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$12\\not \\in P$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$15\\in P$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Penutup kode pertanyaan dan jawaban--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e2. Among the following sets which are subsets of $K=\\{1, 2, 3, 4, 5\\}$ are...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{2, 4, 6}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{1, 3, 5, 7}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e{1, 3, 5}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{0, 2, 4, 6}\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e3. The number of subsets of $P=\\{3, 5, 7, 9, 11\\}$ that have 3 members is...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$16$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$10$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$8$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$4$\u003c/label\u003e\n \u003c/form\u003e\n\u003c!--Tombol Soal, Sesuaikan dengan jumlah soal--\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e4. Among the following sets that are mutually exclusive with the set {m, e, r, a, h} are...\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e{b, i, r, u}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{white}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{black}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{m, e, t, a, l, i, k}\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e5. Given: $P=\\{\\text{factors of 10}\\}$ and $Q=\\{\\text{first three prime numbers}\\}$ then $P\\cup Q=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{1, 2, 3, 4, 5, 7, 10}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{1, 2, 3, 4, 5, 10}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{1, 2, 3, 5, 7, 10}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e{1, 2, 3, 5, 10}\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e6. Given: $A=\\{\\text{factor of 12}\\}$ and $B=\\{\\text{prime number less than 15}\\}$, then $A\\cap B=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e{2, 3}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{2, 3, 4, 6}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{1, 2, 3, 4, 6, 12}\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e{1, 2, 3, 4, 6, 11, 13}\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e7. If the set $A\\subset B$ with $n(A)=11$ and $n(B)=18$, then $n(A\\cap B)=\\cdots$\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$7$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$11$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$18$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$28$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e8. The craftsmen's association has 73 members, 42 of whom produce rattan weaving\n and 37 people produce rattan woven and bamboo woven. a lot of people just\n producing woven bamboo is... people.\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$31$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$36$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$42$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$68$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e9. Of the 30 small children, 14 children like to eat candy and the other 17 children like to eat chocolate. How many children like to eat both?\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$1$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$2$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$3$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$4$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv class=\"box-soal\"\u003e\n\u003cp\u003e10. Of a group of children, 20 children like volleyball, 28 children like basketball, 27 children like table tennis, 13 children like volleyball and basketball, 11 children like volleyball and table tennis, 9 children like basketball and table tennis, and 5 children like all three. How many children don't like these three games, if there are 60 people in the group?\u003c/p\u003e\n \u003cform class=\"jawaban\"\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$10$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$11$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"benar\" /\u003e$12$\u003c/label\u003e\n \u003clabel\u003e\u003cinput name=\"radio\" type=\"radio\" value=\"salah\" /\u003e$13$\u003c/label\u003e\n \u003c/form\u003e\n\u003cdiv id=\"show-hide-box-soal\"\u003e\n\u003cdiv class=\"tombol-show-hide lihat-soal\"\u003e\n closed\n\u003cdiv class=\"menu-tombol-soal\"\u003e\n\u003cdiv class=\"tombol-soal\" data-nomor=\"1\"\u003e1\n\u003cdiv class=\"tombol-soal\" data-nomor=\"2\"\u003e2\n\u003cdiv class=\"tombol-soal\" data-nomor=\"3\"\u003e3\n\u003cdiv class=\"tombol-soal\" data-nomor=\"4\"\u003e4\n\u003cdiv class=\"tombol-soal\" data-nomor=\"5\"\u003e5\n\u003cdiv class=\"tombol-soal\" data-nomor=\"6\"\u003e6\n\u003cdiv class=\"tombol-soal\" data-nomor=\"7\"\u003e7\n\u003cdiv class=\"tombol-soal\" data-nomor=\"8\"\u003e8\n\u003cdiv class=\"tombol-soal\" data-nomor=\"9\"\u003e9\n\u003cdiv class=\"tombol-soal\" data-nomor=\"10\"\u003e10\n\u003cdiv id=\"menu-soal\"\u003e\n\u003cdiv class=\"next-prev-soal\"\u003e\n\u003cdiv class=\"prev\" id=\"prev-soal\"\u003e\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M15.41,16.58L10.83,12L15.41,7.41L14,6L8,12L14,18L15.41,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n Previously\n\u003cdiv class=\"next\" id=\"next-soal\"\u003e\n Furthermore\n \u003csvg viewbox=\"0 0 24 24\"\u003e\n \u003cpath d=\"M8.59,16.58L13.17,12L8.59,7.41L10,6L16,12L10,18L8.59,16.58Z\" fill=\"#000000\"\u003e\u003c/path\u003e\n \u003c/svg\u003e\n\u003cdiv class=\"menu-pilihan-soal\"\u003e\n \u003cbutton class=\"lihat-soal\" id=\"lihat-soal\"\u003eQuestion Menu\u003c/button\u003e\n\u003cdiv class=\"tombol-soal-selesai\"\u003e\n \u003cbutton id=\"soal-selesai\"\u003eFinished\u003c/button\u003e\n\u003cdiv id=\"lihat-hasil-nilai\"\u003e\n\u003cdiv id=\"hasil-pengerjaan\"\u003e\n\u003cp class=\"info-keterangan-hasil\"\u003e*Please screenshot the results and send via WhatsApp **\u003ca href=\"https://www.aantriono.com/p/contact.html\" target=\"_blank\"\u003ehere\u003c/a\u003e\n\u003cp\u003e**, or you can also share it in the comments, and if you remedial you can repeat it again by clicking **\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-himpunan.html\"\u003erepeat\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e**\u003c/p\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eAfter you have finished taking the set material quiz, how did you get your score? successful or need to repeat? Don\u0026rsquo;t hesitate to share it in the comments. That\u0026rsquo;s the article about the set material quiz taught in class 7 of SMP/MTs. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Presenting Set Operations in a Venn Diagram", "date_published" : "2021-09-06T14:57:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menyajikan-operasi-himpunan-dalam-diagram-venn/", "url" : "https://www.aantriono.com/en/menyajikan-operasi-himpunan-dalam-diagram-venn/", "summary": " IntroductionAfter the previous post we discussed meaning of set and presenting relationships between sets with a Venn diagram, so on this occasion we will discuss presenting set operations in Venn diagrams, the properties of set operations, and their application in solving everyday problems related to the concept of sets.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eAfter the previous post we discussed \u003ca href=\"https://www.aantriono.com/2021/08/pengertian-himpunan.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003emeaning of set\u003c/a\u003e and \u003ca href=\"https://www.aantriono.com/2021/08/menyajikan-hubungan-antarhimpunan-dengan-diagram-venn.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003epresenting relationships between sets with a Venn diagram\u003c/a\u003e, so on this occasion we will discuss presenting set operations in Venn diagrams, the properties of set operations, and their application in solving everyday problems related to the concept of sets.\u003c/p\u003e\n\u003ch2 id=\"types-of-set-operations\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-set-operations\" class=\"header-mark\" aria-label=\"Direct link to section: Types of Set Operations\"\u003e\u003c/a\u003eTypes of Set Operations\u003c/h2\u003e\u003cp\u003eHere we will discuss set operations which include:\u003c/p\u003e\n\u003col class=\"style1\"\u003e\n \u003cli\u003eintersection of two sets\u003c/li\u003e\n \u003cli\u003ea combination of two sets\u003c/li\u003e\n \u003cli\u003edifference (difference) of two sets\u003c/li\u003e\n \u003cli\u003ecomplement of a set\u003c/li\u003e\n \u003c/ol\u003e\n\u003ch3 id=\"set-slices\" class=\"headerLink\"\u003e\n \u003ca href=\"#set-slices\" class=\"header-mark\" aria-label=\"Direct link to section: Set Slices\"\u003e\u003c/a\u003eSet Slices\u003c/h3\u003e\u003cp\u003eLook at the following illustration. The results of a questionnaire from a group of students who liked basketball and football showed that Andi, Budi, Candra and Doni liked basketball. Apart from that, it is also noted that Eka, Fahmi, Budi and Doni like football. According to the results of the questionnaire above, are there students who like both types of sports?If presented in set form, the basketball fan set is denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003eAndi, Budi, Candra, Doni\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{\\text{Andi, Budi, Candra, Doni}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eAndi, Budi, Candra, Doni\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the football fan set is denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003eEka, Fahmi, Budi, Doni\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{\\text{Eka, Fahmi, Budi, Doni}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eEka, Fahmi, Budi, Doni\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhakWnBxEAh7z9WyJTpCJosk9EMvU_AFN5s1rBV6c8eSWA60tHrNecU56mc_UkFuMkHBz32p2n3JKZ4hkIM3PlgXX-bVwbHxfpq_doc8Mc97sA171XmOAVahqfOWm2GMn56Kp_bfdNepsZiiRnXuXLmYQ22dcs-1L1QNhYzWXCgwGnvKueeAAJG9LaA/w640-h200/himpunan.png' alt=\"Irisan dua himpunan dengan diagram venn\" \u003e\u003c/p\u003e\n\u003cp\u003eIf the hobbies of a group of students are depicted using a Venn diagram, it appears that Budi and Doni like basketball and football which are intersections of the sets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. This can be written \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003eBudi, Doni\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\\cap B=\\{\\text{Budi, Doni}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eBudi, Doni\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. From this description it can be concluded as follows.\u003c/p\u003e\n\u003cp class=\"note\"\u003e\u003ckbd\u003eConclusion\u003c/kbd\u003eThe intersection of the sets $A$ and $B$ is a set whose members are members of $A$ and $B$, denoted by $A\\cap B=\\{x|x\\in A\\ \\text{and}\\ x\\in B\\}$\u003c/p\u003e\nExample. Given $A=\\{1, 2, 3, 4, 5\\}$ and $B=\\{4, 5, 6, 7\\}$a. Determine $A\\cap B$ by registering members.b. Make a Venn diagram of the problem. Answer.a. $A=\\{1, 2, 3, 4, 5\\}$ and $B=\\{4, 5, 6, 7\\}$There are the same members of $A$ and $B$, namely $4$ and $5$ then $A\\cap B=\\{4, 5\\}$b. The Venn diagram image looks like the following image.\n\u003ctable class=\"tr-caption-container\" style=\"margin-left: auto; margin-right: auto; text-align: center;\"\u003e\n \u003ctbody\u003e\n\u003ctr\u003e\n \u003ctd\u003e\n \u003cimg alt=\"irisan dua himpunan dengan diagram venn\" height=\"279\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp8GuPvweowpHWjdB8beaxBnB_rxjh-u_seZkGDS_noEtWIqYO9oKWW1UW72wvlfPGg39_HHmljQnjuTj9oqcqc2IJSDdVGkDmQ8UUuJDPGbb-2K9eIhKrm4nPA9enNy-D5DWHTvTzg64/w400-h283/photo1630853809+%25284%2529.jpeg\" title=\"Irisan dua himpunan\" width=\"400\" /\u003e\n \u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n \u003ctd class=\"tr-caption\"\u003eImage: Intersection of two sets\u003c/td\u003e\n\u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003ch3 id=\"combination-of-two-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#combination-of-two-sets\" class=\"header-mark\" aria-label=\"Direct link to section: Combination of Two Sets\"\u003e\u003c/a\u003eCombination of Two Sets\u003c/h3\u003e\u003cp\u003eConsider the following two sets.\u003c/p\u003e\n\u003cdiv class=\"rumus\"\u003e$A=\\{\\text{badminton, tennis, basketball, volleyball, football}\\}$\n\u003c/div\u003e\n\u003cdiv class=\"rumus\"\u003e$B=\\{\\text{football, karate, pencak silat, wrestling, judo}\\}$If the members of $A$ and $B$ are combined, a new set will be obtained. The new set is written\n\u003c/div\u003e\n\u003cdiv class=\"rumus\"\u003e$A\\cup B=\\{\\text{badminton, tennis, basketball, volleyball, football, karate, pencak silat, wrestling, judo}\\}$.The combination of these two sets can be expressed using a Venn diagram as shown in the following image.\n\u003c/div\u003e\n\u003ctable class=\"tr-caption-container\" style=\"margin-left: auto; margin-right: auto; text-align: center;\"\u003e\n \u003ctbody\u003e\n\u003ctr\u003e\n \u003ctd\u003e\n \u003cimg alt=\"gabungan dua himpunan dengan diagram venn\" height=\"279\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3dkworLWvWBBVY-F9CBUxaa9vjClkON21aTjIUwyZcOg3UYyZnqTdHFforG7iHBmtebLcooaJMtQ9WwyxFJ8wZnaMCquIB5WxkosGzyGSTM5tM5n0wxVyIKep52HlWNvmd4bEDgnB1TU/w400-h278/photo1630853809+%25282%2529.jpeg\" title=\"Gabungan dua himpunan\" width=\"400\" /\u003e\n \u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n \u003ctd class=\"tr-caption\"\u003eImage: Combination of two sets\u003c/td\u003e\n\u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp class=\"note\"\u003e\u003ckbd\u003eConclusion\u003c/kbd\u003eThe combination of sets $A$ and $B$ is a set whose members are members of set $A$ or members of set $B$, denoted by $A\\cup B=\\{x|x\\in A\\ \\text{or}\\x\\in B\\}$\u003c/p\u003e\nExample. Given $A=\\{2, 4, 6, 8, 10, 12\\}$ and $B=\\{2, 6, 10, 14, 18\\}$a. Determine $A\\cup B$ by listing its members.b. Create a Venn diagramAnswer.a. \\(\\begin{array}{rcl}A\u0026=\u0026\\{2, 4, 6, 8, 10, 12\\}\\\\B\u0026=\u0026\\{2, 6, 10, 14, 18\\}\\\\A\\cup B\u0026=\u0026\\{2, 4, 6, 8, 10, 12, 14, 18\\}\\end{array}\\)b. The Venn diagram looks like the following image.\n\u003ctable class=\"tr-caption-container\" style=\"margin-left: auto; margin-right: auto; text-align: center;\"\u003e\n \u003ctbody\u003e\n\u003ctr\u003e\n \u003ctd\u003e\n \u003cimg alt=\"gabungan dua himpunan dengan diagram venn\" height=\"279\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwPfwyV3iWX5YVuDXqRd4zVheNBa9kHUfmEMlofgMn7jqO3SfZrG8WYolGxUc5MAB_4DTaZo0BmF3c048acyVYE0cJTCBq8jsKd47gPAuAsnhu6btnL3lT1q9DvsO0pmXNY4cYQ3-Ji7g/w400-h279/photo1630853809+%25283%2529.jpeg\" title=\"Gabungan dua himpunan\" width=\"400\" /\u003e\n \u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n \u003ctd class=\"tr-caption\"\u003eImage: Combination of two sets\u003c/td\u003e\n\u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003ch3 id=\"difference-difference-of-two-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#difference-difference-of-two-sets\" class=\"header-mark\" aria-label=\"Direct link to section: Difference (Difference) of Two Sets\"\u003e\u003c/a\u003eDifference (Difference) of Two Sets\u003c/h3\u003e\u003cp\u003eLook at the following two numbers.\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003eprime number less than 15\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{\\text{prime number less than 15}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eprime number less than 15\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cdiv class=\"rumus\"\u003e$B=\\{\\text{odd numbers more than 5 and less than 20}\\}$Are there prime numbers less than 15, but not odd numbers more than 5 and less than 20? If you make a Venn diagram of these two sets, you will get something like the following picture.\n\u003c/div\u003e\n\u003ctable class=\"tr-caption-container\" style=\"margin-left: auto; margin-right: auto; text-align: center;\"\u003e\n \u003ctbody\u003e\n\u003ctr\u003e\n \u003ctd\u003e\n \u003cimg alt=\"selisih dua himpunan dengan diagram venn\" height=\"279\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinTRWkb66hx36iIov14k9PBrL0r2k5iBIqXQQSP8SxwH6CDwP2SfxMDLxRTOjsIo-fkXKeAZfshzmWSgF094sXNnrLAmjETv3KtnkH56kArCc2AtmkChrpqslLk5a7Zk9HNSqHgC_CBoU/w400-h286/photo1630853809.jpeg\" title=\"Selisih dua himpunan\" width=\"400\" /\u003e\n \u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n \u003ctd class=\"tr-caption\"\u003eFigure: Difference between two sets\u003c/td\u003e\n\u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003ePrime numbers that are less than 15 but not odd numbers that are more than 5 and less than 20 are the shaded part of the figure. In the shaded part there are the numbers 2, 3, and 5. The numbers 2, 3, and 5 are members of the set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, but not members of the set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Next 2, 3, and 5 are called the differences of the sets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp class=\"note\"\u003e\u003ckbd\u003eConclusion\u003c/kbd\u003eThe difference between sets $A$ and $B$ is a set whose members are members of set $A$, but not members of set $B$, denoted by $A-B=\\{x|x\\in A, x\\not\\in B\\}$\u003c/p\u003e\n\u003cp\u003eExample. Given\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ee\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{a, b, c, d, e, f\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ee\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ee\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eo\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{a, e, i, o, u\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ee\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eo\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eu\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003ea. Determine \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA-B\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e by listing its members.b. Create a Venn diagram and shade \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA-B\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e Answer.\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"right center left\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ee\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ee\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eo\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{array}{rcl}a.\\ A\u0026amp;=\u0026amp;\\{a, b, c, d, e, f\\}\\\\B\u0026amp;=\u0026amp;\\{a, e, i, o, u\\}\\\\A-B\u0026amp;=\u0026amp;\\{b, c, d, f\\}\\end{array}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.6em;vertical-align:-1.55em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e.\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ee\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ee\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ei\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eo\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eu\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eb. Venn diagram\u003c/p\u003e\n\u003ctable class=\"tr-caption-container\" style=\"margin-left: auto; margin-right: auto; text-align: center;\"\u003e\n \u003ctbody\u003e\n\u003ctr\u003e\n \u003ctd\u003e\n \u003cimg alt=\"selisih dua himpunan dengan diagram venn\" height=\"279\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrdqiq7vObsEsog1Gcn115G5DgF8023oG0RiBnHKA_NuzzCGpMdyRh5hYtS58tYaDIpUCYYtUqO6rZodAi4_jVYoKN_7QMFPOw0E1yH9DqAhGgNB7t8w1IDTu7cvdxhLdTuQgNr59Zsgw/w400-h270/photo1630894013.jpeg\" title=\"Selisih dua himpunan\" width=\"400\" /\u003e\n \u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n \u003ctd class=\"tr-caption\"\u003eFigure: Difference between two sets\u003c/td\u003e\n\u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003ch3 id=\"complement-of-a-set\" class=\"headerLink\"\u003e\n \u003ca href=\"#complement-of-a-set\" class=\"header-mark\" aria-label=\"Direct link to section: Complement of a Set\"\u003e\u003c/a\u003eComplement of a Set\u003c/h3\u003e\u003cp\u003eLook at the following Venn diagram.\u003c/p\u003e\n\u003ctable class=\"tr-caption-container\" style=\"margin-left: auto; margin-right: auto; text-align: center;\"\u003e\n \u003ctbody\u003e\n\u003ctr\u003e\n \u003ctd\u003e\n \u003cimg alt=\"komplemen suatu himpunan dengan diagram venn\" height=\"279\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQskS5YxqGyE94gWpcnx7fkf81Eqcu0ySD8F5az-KFAEEpRT5fUslCKc-Aun_kHo_wEkNKxc2LUhazAwQusN0YLRwAhVuI6YoEyQfL43YXMIzohpOrcRDdgIKzIHgkfsp5rRG2WOONUys/w400-h309/photo1630853809+%25285%2529.jpeg\" title=\"Komplemen suatu himpunan\" width=\"400\" /\u003e\n \u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n \u003ctd class=\"tr-caption\"\u003eFigure: Complement of a set\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\n\u003cp\u003eThe Venn diagram shows \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS=\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{1, 3, 5, 7\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eIs there a set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e that is not a member of the set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e? What is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e⊂\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\\subset S\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⊂\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e ? The members of the set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e that are not members of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are called \u003cem\u003ecomplement or complement of\u003c/em\u003e \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. The complement of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is written \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA^c\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6644em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Based on the Venn diagram in the image above, we get \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;=A^c=\\{2, 4, 6, 8, 9, 10\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6644em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.Thus, the definition of complement of a set is as follows.\u003c/p\u003e\n\u003cp class=\"note\"\u003e\u003ckbd\u003eConclusion\u003c/kbd\u003eIf $A$ is a set in $S$ then the members of the set $S$ that are not members of $A$ are called *complement* $A$ and written $A'$ or $A^c$, denoted by $A'=A^c=\\{x|x\\in S\\ \\text{and}\\ x\\not\\in A\\}$\u003c/p\u003e\n\u003cp\u003eExample. For example:\u003c/p\u003e\n\u003cdiv class=\"rumus\"\u003e\\(\\begin {array}{rcl} S\u0026=\u0026\\{\\text{set\\ names\\ months\\ in\\ one\\ year}\\}\\\\A\u0026=\u0026\\{\\text{January, February, May, June, July}\\}\\\\B\u0026=\u0026\\{\\text{September, October, November, December}\\}\\end{array}\\)Determine:a. $A'$ by naming its members.b. $B'$ by naming its members. Answer.a. The members of $S$ who are not members of $A$ are March, April, August, September, October, November, and December. So :\n\u003c/div\u003e\n\u003cdiv class=\"rumus\"\u003e$A'=\\{\\text{March, April, August, September, October, November, December}\\}$b. The members of $S$ who are not members of $B$ are January, February, March, April, May, June, July, and August. So :\n\u003c/div\u003e\n\u003cdiv class=\"rumus\"\u003e$B'=\\{\\text{January, February, March, April, May, June, July, August}\\}$\n## Properties of Set Operations\u003c/div\u003e\nThere are several properties that need to be known when operating two or more sets. Suppose we know the following sets.\n\\(\\begin {array}{rcl} S\u0026=\u0026\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}\\\\A\u0026=\u0026\\{1, 2, 3, 4, 5, 6, 7, 8\\}\\\\B\u0026=\u0026\\{3, 4, 5, 6\\}\\\\C\u0026=\u0026\\{3, 4, 5, 10\\}\\end{array}\\)\nUsing these sets, let's test some properties of sets.\n### Commutative Property\nConsider the following set operations.\n$A=\\{1, 2, 3, 4, 5, 6, 7, 8\\}$\n$B=\\{3, 4, 5, 6\\}$\n\\(\\begin {array}{rcl} A\\cup B\u0026=\u0026\\{1, 2, 3, 4, 5, 6, 7, 8\\}\\\\B\\cup A\u0026=\u0026\\{1, 2, 3, 4, 5, 6, 7, 8\\}\\\\A\\cap B\u0026=\u0026\\{3, 4, 5, 6\\}\\\\B\\cap A\u0026=\u0026\\{3, 4, 5, 6\\}\\end{array}\\)\n#### Notes\n* The commutative property of slices is: $A\\cap B=B\\cap A$\n* The commutative property of combination is: $A\\cup B=B\\cup A$\n### Associative Properties\nLook at the following sets.\n$A=\\{1, 2, 3, 4, 5, 6, 7, 8\\}$\n$B=\\{3, 4, 5, 6\\}$\n$C=\\{3, 4, 5, 10\\}$\nThen we will perform the intersection operation of these sets.\n\\(\\begin {array}{rcl} A\\cap B\u0026=\u0026\\{3, 4, 5, 6\\}\\\\B\\cap C\u0026=\u0026\\{3, 4, 5\\}\\\\(A\\cap B)\\cap C\u0026=\u0026\\{3, 4, 5\\}\\\\A\\cap (B\\cap C)\u0026=\u0026\\{3, 4, 5\\}\\end{array}\\)\nNext is the operation to combine these sets, as shown below.\n\\(\\begin {array}{rcl} A\\cup B\u0026=\u0026\\{1, 2, 3, 4, 5, 6, 7, 8\\}\\\\B\\cup C\u0026=\u0026\\{3, 4, 5, 6, 10\\}\\\\(A\\cup B)\\cup C\u0026=\u0026\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}\\\\A\\cup (B\\cup C)\u0026=\u0026\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}\\end{array}\\)\nFrom the description above, it can be concluded as follows.\n\u003cp class=\"note\"\u003e\u003ckbd\u003eConclusion\u003c/kbd\u003eFor any set $A, B\\ \\text{and}\\ C$ the following properties apply:\n\\(\\begin{array}{rcl}(A\\cap B)\\cap C\u0026=\u0026A\\cap (B\\cap C)\\\\(A\\cup B)\\cup C\u0026=\u0026A\\cup (B\\cup C)\\end{array}\\)\u003c/p\u003e\n\u003ch3 id=\"distributive-property\" class=\"headerLink\"\u003e\n \u003ca href=\"#distributive-property\" class=\"header-mark\" aria-label=\"Direct link to section: Distributive Property\"\u003e\u003c/a\u003eDistributive Property\u003c/h3\u003e\u003cp\u003eLook at the following sets.\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{1, 2, 3, 4, 5, 6, 7, 8\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{3, 4, 5, 6\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC=\\{3, 4, 5, 10\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\nThen we will perform intersection and combination operations on these sets.\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"right center left\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e∪\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin {array}{rcl} A\\cap B\u0026amp;=\u0026amp;\\{3, 4, 5, 6\\}\\\\A\\cap C\u0026amp;=\u0026amp;\\{3, 4, 5\\}\\\\(A\\cap B)\\cup (A\\cap C)\u0026amp;=\u0026amp;\\{3, 4, 5, 6\\}\\end{array}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.6em;vertical-align:-1.55em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∪\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan 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style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan 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class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.95em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\nFrom the description above, it can be concluded as follows.\u003c/p\u003e\n\u003cp class=\"note\"\u003e\u003ckbd\u003eConclusion\u003c/kbd\u003eFor any set $A, B, \\text{and}\\ C$ the following properties apply:\n\\(\\begin{array}{rcl}A\\cap (B\\cup C)\u0026=\u0026(A\\cap B)\\cup (A\\cap C)\\\\A\\cup (B\\cap C)\u0026=\u0026(A\\cup B)\\cap (A\\cup C)\\end{array}\\)\u003c/p\u003e\n\u003ch2 id=\"application-of-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#application-of-sets\" class=\"header-mark\" aria-label=\"Direct link to section: Application of Sets\"\u003e\u003c/a\u003eApplication of Sets\u003c/h2\u003e\u003cp\u003eTo solve everyday problems related to the concept of sets, it will be easier to use Venn diagrams to solve them. Consider the following examples.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eThe number of students in class VIIA is 32 people, 20 of whom like mathematics and 14 people like English. a. Create a Venn diagram of the problem. b. How many students like mathematics and English?\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIn a class there are 17 students who take extra-curricular Rohis, 15 students take PMR, and 8 students take both.a. Create a Venn diagram of the problem.b. How many students in total?\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIn selecting scholarship recipients, each student must pass the mathematics and language tests. Of the 180 participants, 103 people were declared to have passed the mathematics test and 142 people were declared to have passed the language test. a. Create a Venn diagram of the problem.b. How many students are declared to have graduated as scholarship recipients?\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eOf the 80 students who were surveyed about their hobby of watching sports on television, data was obtained that 48 people liked watching volleyball, 42 people liked watching basketball and 10 people did not like watching this event. Many students who like watching volleyball and basketball are\u0026hellip;Answer.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003ea. Venn diagram\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFctKbLiAzo81SgLtkfsEDTk9kJxT6TpC_dhCBMtNwAvouxCfIO9NNRhx9YyNwRLe7yQ-b181gmneIhmI7pUPr5Z5ydF1ed3bxYWvYGiPK2qZ6z_hYZO_t0iK8KzufQ1xJcMWantlqPT0/w400-h278/oh8.png' alt=\"Irisan himpunan di diagram Venn\" \u003e\u003c/p\u003e\n\u003cp\u003eb. For example, students who like mathematics and English are \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"right center left\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e32\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e32\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e32\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e34\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e32\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e34\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e32\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{array}{rcl}(20-x)+x+14-x\u0026amp;=\u0026amp;32\\\\20-x+x+14-x\u0026amp;=\u0026amp;32\\\\20+14-x\u0026amp;=\u0026amp;32\\\\34-x\u0026amp;=\u0026amp;32\\\\x\u0026amp;=\u0026amp;34-32\\\\x\u0026amp;=\u0026amp;2\\end{array}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.2em;vertical-align:-3.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.85em;\"\u003e\u003cspan style=\"top:-6.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e34\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.35em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.85em;\"\u003e\u003cspan style=\"top:-6.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.35em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.85em;\"\u003e\u003cspan style=\"top:-6.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e34\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.35em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eSo, there are 2 students who like mathematics and English.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003ea. Venn diagram\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqaxj5PFt26LbfZtjWkCIh_hHFO1hyphenhyphenWtVhsxi4KOt3HWQPL-heBb-9KiHarUDvxiF0M2plo6mYO6VdIjPgMQr-KdjOt9UiOagQ4uF3xUbzNWmm9aztSYzx4QPepmZ_y4ny_0L_YvVmpas/w400-h279/oh9.png' alt=\"irisan himpunan di diagram Venn\" \u003e\u003c/p\u003e\n\u003cp\u003eb. For example:\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003estudents attending spiritual\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eR=\\{\\text{students attending spiritual}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003estudents attending spiritual\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e17\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(R)=17\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e17\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003estudents taking PMR\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP=\\{\\text{students taking PMR}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003estudents taking PMR\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(P)=15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003estudents who attend rohis and PMR\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eR\\cap P=\\{\\text{students who attend rohis and PMR}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003estudents who attend rohis and PMR\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(R\\cap P)=8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eMany of the students in the class were:\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"right center left\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e∪\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e17\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{array}{rcl}n(R\\cup P)\u0026amp;=\u0026amp;n(R)+n(P)-n(R\\cap P)\\\\\u0026amp;=\u0026amp;17+15-8\\\\\u0026amp;=\u0026amp;24\\end{array}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.6em;vertical-align:-1.55em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∪\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.05em;\"\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e17\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.55em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e3. a. Venn diagram\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-EubWI2tQpwuBMJ_yVIh4eegrL8LM_BTPFELsVu-LUUj2edJlDYLeSqbEiaT3FDTYYCi2R3UuXg9ZLOEWCXoKepwoYrKRptg6WqQuF1fKT70O18TaAR366kNAHBgkV5KC44Hlc87x9E8/w400-h278/oh10.png' alt=\"irisan himpunan di diagram Venn\" \u003e\u003c/p\u003e\n\u003cp\u003eb. For example:\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003estudents who passed mathematics\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eM=\\{\\text{students who passed mathematics}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003estudents who passed mathematics\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e103\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(M)=103\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e103\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003elanguage pass students\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{\\text{language pass students}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003elanguage pass students\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e142\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(B)=142\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e142\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003etest participants\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS=\\{\\text{test participants}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003etest participants\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(S)=180\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003estudents who passed the selection\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eM\\cap B=\\{\\text{students who passed the selection}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003estudents who passed the selection\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(M\\cap B)=x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eLook at the Venn diagram above.\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"right center left\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e103\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e142\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e103\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e142\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e245\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e245\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e65\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{array}{rcl}n(S)\u0026amp;=\u0026amp;(103-x)+(x)+(142-x)\\\\180\u0026amp;=\u0026amp;103+142-x\\\\180\u0026amp;=\u0026amp;245-x\\\\x\u0026amp;=\u0026amp;245-180\\\\x\u0026amp;=\u0026amp;65\\end{array}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e103\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e142\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e103\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e142\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e245\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e245\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e65\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eSo \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e65\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eM\\cap B=x=65\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e65\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eThis means that the number of students receiving scholarships is 65 people.\u003ckbd\u003eAnother way :\u003c/kbd\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∪\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(S)=n(M\\cup B)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∪\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"right center left\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e103\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e142\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e245\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e245\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e65\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{array}{rcl}n(S)\u0026amp;=\u0026amp;n(M)+n(B)-n(M\\cap B)\\\\180\u0026amp;=\u0026amp;103+142-n(M\\cap B)\\\\180\u0026amp;=\u0026amp;245-n(M\\cap B)\\\\n(M\\cap B)\u0026amp;=\u0026amp;245-180\\\\\u0026amp;=\u0026amp;65\\end{array}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e103\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e142\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e245\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e245\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e65\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"4\"\u003e\n\u003cli\u003eFor example:\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003elikes watching volleyball\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eV=\\{\\text{likes watching volleyball}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003elikes watching volleyball\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e48\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(V)=48\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e48\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003elikes watching basketball\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{\\text{likes watching basketball}\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003elikes watching basketball\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e42\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(B)=42\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e42\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003edoesn’t like watching volleyball and basketball\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eV\u0026#x27;\\cap B\u0026#x27;=\\{\\text{doesn\u0026#x27;t like watching 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mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003edoesn’t like watching volleyball and basketball\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(V\u0026#x27;\\cap B\u0026#x27;=10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eSo :\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"right center left\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e∪\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e80\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e48\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e42\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e80\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e80\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e∩\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmo lspace=\"0em\" rspace=\"0em\"\u003e=\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{array}{rcl}n(V\\cup B)\u0026amp;=\u0026amp;n(V)+n(B)-n(V\\cap B)+n(V\u0026#x27;\\cap B\u0026#x27;)\\\\80\u0026amp;=\u0026amp;48+42-n(V\\cap B)+10\\\\80\u0026amp;=\u0026amp;100-n(V\\cap B)\\\\n(V\\cap B)\u0026amp;=\u0026amp;100-80\\\\n(V\\cap B)\u0026amp;=\u0026amp;20\\end{array}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∪\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e80\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" 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style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.21em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e48\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e42\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e∩\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.81em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e80\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\nSo the number of students who like watching volleyball and basketball is 20 people.\u003c/p\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eAfter studying material about set operations, the properties of set operations, and the application of sets in solving everyday problems, now we will carry out an evaluation to measure understanding of the material. Please take the following quiz by clicking the button below.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://www.aantriono.com/2021/09/kuis-materi-himpunan.html\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eQuiz\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThus, the article on set material in sub-material presents set operations in Venn diagrams, as well as the properties of set operations, and the application of the set concept in solving everyday problems related to the set concept. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Presenting Relationships Between Sets Using Venn Diagrams", "date_published" : "2021-08-30T04:28:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menyajikan-hubungan-antarhimpunan-dengan-diagram-venn/", "url" : "https://www.aantriono.com/en/menyajikan-hubungan-antarhimpunan-dengan-diagram-venn/", "summary": " IntroductionSets can be presented in several ways:\nby description, by set-builder notation, by listing their elements, or by using a Venn diagram. This diagram is named after the British mathematician John Venn (1834-1923).\nBasic guidelines for drawing a Venn diagram:\nThe universal set SSS is drawn as a rectangle. Each set inside the universal set is shown by a simple closed curve. Each member of a set is represented by a point or written inside its region. If a set has many members, the members do not have to be written one by one. Example of a Venn diagram:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eSets can be presented in several ways:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eby description,\u003c/li\u003e\n\u003cli\u003eby set-builder notation,\u003c/li\u003e\n\u003cli\u003eby listing their elements,\u003c/li\u003e\n\u003cli\u003eor by using a \u003cstrong\u003eVenn diagram\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThis diagram is named after the British mathematician \u003cstrong\u003eJohn Venn\u003c/strong\u003e (1834-1923).\u003c/p\u003e\n\u003cp\u003eBasic guidelines for drawing a Venn diagram:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eThe universal set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is drawn as a rectangle.\u003c/li\u003e\n\u003cli\u003eEach set inside the universal set is shown by a simple closed curve.\u003c/li\u003e\n\u003cli\u003eEach member of a set is represented by a point or written inside its region.\u003c/li\u003e\n\u003cli\u003eIf a set has many members, the members do not have to be written one by one.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eExample of a Venn diagram:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9281HaalIJ4N4tU1cbSFdISOF5BiBlSrocw-MVM0rnVatQFcvWkROqf1gF_R6JDLHXQUfmvj0TDNGGn6_Wb1Zn9W-VCVcw-1ZP6AR5SIq06lc78Ecb_MPk8Ofq2kPIUYA8gPoQ3-1CJMlAJxzO6bBWaEGl6m0bkX36z8HeAC5XVAohvz_OFBYrogq/w400-h125/himpunan1.png' alt=\"Example of a Venn diagram\" \u003e\u003c/p\u003e\n\u003ch2 id=\"relationships-between-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#relationships-between-sets\" class=\"header-mark\" aria-label=\"Direct link to section: Relationships Between Sets\"\u003e\u003c/a\u003eRelationships Between Sets\u003c/h2\u003e\u003cp\u003eDifferent relationships between sets can be shown with the following Venn diagrams.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEnIWHdotC_jUMNMAzGMaeLpFw-qx1IbIN1fi8ypbVnciCwraEkqNpZ4MAqPum0ds6eo9sb8m63Uzi48zL2gGPVwEGv_ikfZiHjOa_5dTnMCaP_FdcW_n_LdZdhaZCLUF7FsRGXWJk2nk/w400-h356/himp1.png' alt=\"Relationships between sets\" \u003e\u003c/p\u003e\n\u003ch3 id=\"1-subsets\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-subsets\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Subsets\"\u003e\u003c/a\u003e1. Subsets\u003c/h3\u003e\u003cp\u003eIn figure (a), set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a subset of set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e⊂\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA \\subset B\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⊂\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThis means every element of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is also an element of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"2-equal-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-equal-sets\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Equal Sets\"\u003e\u003c/a\u003e2. Equal Sets\u003c/h3\u003e\u003cp\u003eIn figure (b), every element of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is an element of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and every element of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is an element of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = B\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-disjoint-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-disjoint-sets\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Disjoint Sets\"\u003e\u003c/a\u003e3. Disjoint Sets\u003c/h3\u003e\u003cp\u003eIn figure (c), sets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e have no common elements, so they are called \u003cstrong\u003edisjoint sets\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"4-intersecting-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-intersecting-sets\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Intersecting Sets\"\u003e\u003c/a\u003e4. Intersecting Sets\u003c/h3\u003e\u003cp\u003eIn figure (d), sets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e have common elements, but each also has elements not shared by the other.\u003c/p\u003e\n\u003ch2 id=\"equivalent-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#equivalent-sets\" class=\"header-mark\" aria-label=\"Direct link to section: Equivalent Sets\"\u003e\u003c/a\u003eEquivalent Sets\u003c/h2\u003e\u003cp\u003eTo understand equivalent sets, consider the following example.\u003c/p\u003e\n\u003cp\u003eIf:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny = x + 1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e belonging to the whole numbers set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e belonging to the natural numbers set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eN\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then we obtain:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e0\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e1\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e2\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e3\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e\u0026hellip;\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eFrom the table:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nC = \\{0,1,2,3,\\cdots\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nN = \\{1,2,3,4,\\cdots\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe relationship between \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eN\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e can be shown as a one-to-one correspondence.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbrgNmu-q9IiJV0DxQ2M_ZDaCCn1MuggTWR6rogt4rbWxple9FC4XM-IlQwsE7Q2cmYuRBcujKxXm8HEV8NiayH3IzWKLtqZkZitsW-Kd6KDsw7T9B6lpHeDZZaz70URMRQa4bSWeMQJg/w320-h307/Screenshot+2021-08-30+11.18.50.png' alt=\"Arrow diagram\" \u003e\u003c/p\u003e\n\u003cp\u003eBecause both sets have the same number of elements, they are \u003cstrong\u003eequivalent sets\u003c/strong\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e∼\u003c/mo\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nC \\sim N\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∼\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eConclusion:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eTwo sets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are said to be equivalent if \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(A)=n(B)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, written as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e∼\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA \\sim B\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∼\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003c/blockquote\u003e\u003ch2 id=\"example-problems\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problems\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problems\"\u003e\u003c/a\u003eExample Problems\u003c/h2\u003e\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{1,2,3,4,5,6,7,8,9,10\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = \\{1,3,5,7\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB = \\{2,3,5,7,9\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the universal set, draw a Venn diagram for these sets.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH0lr1V6P7AqAMclhZ8Vp_K-NFnq38TzC8oe1ijIB1Us8LszDdz_z2FVJ7IDDhS3g1hvL9VyJz0XLk0S4fpZgxgq4F6jdCyWk8JnD8Uu0JnK2d8jFIz3ZRpBSQrD2zEtmx5bZSSMVi-qg/w400-h300/himp3.png' alt=\"Venn diagram example 1\" \u003e\u003c/p\u003e\n\u003cp\u003eIn the diagram, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the common elements of sets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{0,1,2,3,\\cdots,10\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = \\{1,2,3,4\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB = \\{5,6,7,8\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nC = \\{0,1,2,3\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nD = \\{2,3,4\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nE = \\{2,3,5,7\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nF = \\{3,5,7\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eQuestions:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eShow the pairs of sets with Venn diagrams.\u003c/li\u003e\n\u003cli\u003eWhich pairs are disjoint, intersecting, and subsets?\u003c/li\u003e\n\u003cli\u003eAre there any equal sets?\u003c/li\u003e\n\u003cli\u003eAre there any equivalent sets?\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs2O9rrYqWPbvjYTnmqYLHmf9BcYUP8yhIprYSHOhPRjL9yd39_EcDAan4bsefAQ2dVIQ9ipZZIkXAQfLikKj02UMivoLUfbVmNQkzzYWHZJLDar54acIYkPEGs20lY9W9YwVlT6C4pVo/w400-h126/himp2.png' alt=\"Venn diagram example 2\" \u003e\u003c/p\u003e\n\u003cp\u003eSolution:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eSets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are disjoint.\u003c/li\u003e\n\u003cli\u003eSets \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e intersect, with common elements \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eSet \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a subset of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eThere are no equal pairs of sets.\u003c/li\u003e\n\u003cli\u003eThere is an equivalent pair, namely \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, because:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nn(A)=n(B)=4\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eVenn diagrams are an effective way to present several sets at once and show their relationships clearly. With them, subsets, intersections, disjoint sets, and equivalent sets become easier to see and understand.\u003c/p\u003e\n", "language": "en" }, { "title" : "Writing Mathematical Formulas on Blogs", "date_published" : "2021-08-23T15:13:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menulis-rumus-matematika-di-blog/", "url" : "https://www.aantriono.com/en/menulis-rumus-matematika-di-blog/", "summary": " IntroductionEducational blogs, especially those about mathematics or accounting, almost always need symbols, notation, formulas, and equations. The problem is that Blogger does not provide a built-in equation editor like Microsoft Word.\nBecause of that, mathematical formulas on blogs are usually written in LaTeX\\LaTeXLATE​X and then rendered by JavaScript libraries such as MathJax or KaTeX.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eEducational blogs, especially those about mathematics or accounting, almost always need symbols, notation, formulas, and equations. The problem is that Blogger does not provide a built-in equation editor like Microsoft Word.\u003c/p\u003e\n\u003cp\u003eBecause of that, mathematical formulas on blogs are usually written in \u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/strong\u003e and then rendered by JavaScript libraries such as \u003cstrong\u003eMathJax\u003c/strong\u003e or \u003cstrong\u003eKaTeX\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwXhXh-29qqCCQvxftZ0xkRT7jAJsZE-X4IiAoC3Sry_JeDe-LX6VTDExozqLFCkHhix4xu8MuSEmCiQO4f8A7o4aMUzjy-l_cAyk-b1dh_kJHNCoQeWnoS_0OUWmkLl-rstyFoJ-pqaE/w400-h400/20210823_153204.jpg\" title=\"Writing mathematical formulas on a blog\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwXhXh-29qqCCQvxftZ0xkRT7jAJsZE-X4IiAoC3Sry_JeDe-LX6VTDExozqLFCkHhix4xu8MuSEmCiQO4f8A7o4aMUzjy-l_cAyk-b1dh_kJHNCoQeWnoS_0OUWmkLl-rstyFoJ-pqaE/w400-h400/20210823_153204.jpg\" data-sub-html=\"\u003ch2\u003eWriting mathematical formulas on a blog\u003c/h2\u003e\u003cp\u003eWriting mathematical formulas on a blog\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwXhXh-29qqCCQvxftZ0xkRT7jAJsZE-X4IiAoC3Sry_JeDe-LX6VTDExozqLFCkHhix4xu8MuSEmCiQO4f8A7o4aMUzjy-l_cAyk-b1dh_kJHNCoQeWnoS_0OUWmkLl-rstyFoJ-pqaE/w400-h400/20210823_153204.jpg' alt=\"Writing mathematical formulas on a blog\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eWriting mathematical formulas on a blog\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwXhXh-29qqCCQvxftZ0xkRT7jAJsZE-X4IiAoC3Sry_JeDe-LX6VTDExozqLFCkHhix4xu8MuSEmCiQO4f8A7o4aMUzjy-l_cAyk-b1dh_kJHNCoQeWnoS_0OUWmkLl-rstyFoJ-pqaE/s2048/20210823_153204.jpg\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwXhXh-29qqCCQvxftZ0xkRT7jAJsZE-X4IiAoC3Sry_JeDe-LX6VTDExozqLFCkHhix4xu8MuSEmCiQO4f8A7o4aMUzjy-l_cAyk-b1dh_kJHNCoQeWnoS_0OUWmkLl-rstyFoJ-pqaE/s2048/20210823_153204.jpg\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eInstead of inserting formulas as screenshots, writing them directly with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e produces cleaner output and follows mathematical formatting much better.\u003c/p\u003e\n\u003ch2 id=\"general-workflow\" class=\"headerLink\"\u003e\n \u003ca href=\"#general-workflow\" class=\"header-mark\" aria-label=\"Direct link to section: General Workflow\"\u003e\u003c/a\u003eGeneral Workflow\u003c/h2\u003e\u003cp\u003eThe workflow is simple:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ewrite the formula in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003epaste that code into the blog article,\u003c/li\u003e\n\u003cli\u003elet MathJax or KaTeX render it.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"writing-latex-code\" class=\"headerLink\"\u003e\n \u003ca href=\"#writing-latex-code\" class=\"header-mark\" aria-label=\"Direct link to section: Writing LaTeX Code\"\u003e\u003c/a\u003eWriting LaTeX Code\u003c/h3\u003e\u003cp\u003eIf you are not yet comfortable typing raw \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, you can use \u003ca href=\"https://www.codecogs.com/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eCodeCogs\u003c/a\u003e as a helper. It provides a visual formula editor and generates the corresponding \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e code.\u003c/p\u003e\n\u003cp\u003eCodeCogs usually shows three main parts:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ea symbol toolbar,\u003c/li\u003e\n\u003cli\u003ea \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e input box,\u003c/li\u003e\n\u003cli\u003ea preview area.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-1\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\frac\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ex\u003cspan class=\"nb\"\u003e^\u003c/span\u003e2+1\u003cspan class=\"nb\"\u003e}{\u003c/span\u003ex-3\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eRendered result:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{x^2+1}{x-3}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2604em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eWhen copying from CodeCogs, it is better to copy the \u003cstrong\u003eraw \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e code\u003c/strong\u003e instead of the image output. Then paste it into the blog in HTML mode or any editor mode that preserves backslashes correctly.\u003c/p\u003e\n\u003ch2 id=\"common-latex-examples\" class=\"headerLink\"\u003e\n \u003ca href=\"#common-latex-examples\" class=\"header-mark\" aria-label=\"Direct link to section: Common LaTeX Examples\"\u003e\u003c/a\u003eCommon LaTeX Examples\u003c/h2\u003e\u003ch3 id=\"basic-arithmetic\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-arithmetic\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Arithmetic\"\u003e\u003c/a\u003eBasic Arithmetic\u003c/h3\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eLaTeX Code\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eResult\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ea+b\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea+b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ec-d\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec-d\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ea\\times b\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\\times b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ec\\div d\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e÷\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\\div d\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e÷\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"powers-and-subscripts\" class=\"headerLink\"\u003e\n \u003ca href=\"#powers-and-subscripts\" class=\"header-mark\" aria-label=\"Direct link to section: Powers and Subscripts\"\u003e\u003c/a\u003ePowers and Subscripts\u003c/h3\u003e\u003cp\u003eUse \u003ccode\u003e^\u003c/code\u003e for exponents and \u003ccode\u003e_\u003c/code\u003e for subscripts. If the exponent or subscript contains more than one character, wrap it in curly braces.\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eLaTeX Code\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eResult\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ea^b\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea^b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8491em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8491em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ea^{bc}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea^{bc}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8491em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8491em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e4x_a\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4x_a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7944em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ex_{max}^2+x_{min}^2=1\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msubsup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsubsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msubsup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex_{max}^2+x_{min}^2=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0611em;vertical-align:-0.247em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-2.453em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ema\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.247em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0728em;vertical-align:-0.2587em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-2.4413em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003emin\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2587em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"fractions\" class=\"headerLink\"\u003e\n \u003ca href=\"#fractions\" class=\"header-mark\" aria-label=\"Direct link to section: Fractions\"\u003e\u003c/a\u003eFractions\u003c/h3\u003e\u003cp\u003eUse \u003ccode\u003e\\frac\u003c/code\u003e for regular fractions and \u003ccode\u003e\\dfrac\u003c/code\u003e for a larger display style.\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eLaTeX Code\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eResult\u003c/th\u003e\n \u003c/tr\u003e\n 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class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6954em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\dfrac{c}{d}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{c}{d}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\frac{x-x_1^2}{a^2}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsubsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msubsup\u003e\u003c/mrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\frac{x-x_1^2}{a^2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.4791em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1341em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7463em;\"\u003e\u003cspan style=\"top:-2.786em;margin-right:0.0714em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.5102em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8913em;\"\u003e\u003cspan style=\"top:-2.214em;margin-left:0em;margin-right:0.0714em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.931em;margin-right:0.0714em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.286em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003eF=k\\dfrac{Q_1Q_2}{R^2}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmsup\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eF=k\\dfrac{Q_1Q_2}{R^2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n 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stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey=\\int (x+1)\\ dx\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1111em;vertical-align:-0.3061em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0006em;\"\u003e∫\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ey=\\int_{0}^{\\infty} f(x)\\ dx\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsubsup\u003e\u003cmo\u003e∫\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e∞\u003c/mi\u003e\u003c/msubsup\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey=\\int_{0}^{\\infty} f(x)\\ dx\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2151em;vertical-align:-0.3558em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003e\u003cspan class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0006em;\"\u003e∫\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8593em;\"\u003e\u003cspan style=\"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2579em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e∞\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3558em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\iint f(x,y)\\ dx\\ dy\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e∬\u003c/mo\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\iint f(x,y)\\ dx\\ dy\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.111em;vertical-align:-0.306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0005em;\"\u003e∬\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\oint\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e∮\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\oint\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1111em;vertical-align:-0.3061em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop op-symbol small-op\" style=\"margin-right:0.19445em;position:relative;top:-0.0006em;\"\u003e∮\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"derivatives\" class=\"headerLink\"\u003e\n \u003ca href=\"#derivatives\" class=\"header-mark\" aria-label=\"Direct link to section: Derivatives\"\u003e\u003c/a\u003eDerivatives\u003c/h3\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eLaTeX Code\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eResult\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\dfrac{dy}{dx}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{dy}{dx}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\dfrac{\\partial z}{\\partial x}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∂\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∂\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{\\partial z}{\\partial x}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\" style=\"margin-right:0.05556em;\"\u003e∂\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\" style=\"margin-right:0.05556em;\"\u003e∂\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.04398em;\"\u003ez\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"roots-and-inequalities\" class=\"headerLink\"\u003e\n \u003ca href=\"#roots-and-inequalities\" class=\"header-mark\" aria-label=\"Direct link to section: Roots and Inequalities\"\u003e\u003c/a\u003eRoots and Inequalities\u003c/h3\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eLaTeX Code\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eResult\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\sqrt{a}\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{a}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.2397em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8003em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.7603em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.7603em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4141em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.4em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:2.48em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 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xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \u0026lt; b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ea \\le b\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\le b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7719em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ea \u0026gt; b\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \u0026gt; b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003ea \\ge b\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e≥\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\ge b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7719em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"sigma-and-pi-notation\" class=\"headerLink\"\u003e\n \u003ca href=\"#sigma-and-pi-notation\" class=\"header-mark\" aria-label=\"Direct link to section: Sigma and Pi Notation\"\u003e\u003c/a\u003eSigma and Pi Notation\u003c/h3\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eLaTeX Code\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eResult\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\sum_{i=1}^{n} x_i\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmo\u003e∑\u003c/mo\u003e\u003cmrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msubsup\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sum_{i=1}^{n} x_i\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.104em;vertical-align:-0.2997em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003e\u003cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\"\u003e∑\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8043em;\"\u003e\u003cspan style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003cspan class=\"mrel mtight\"\u003e=\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2997em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\sum \\limits_{i=1}^{n} x_i\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmo\u003e∑\u003c/mo\u003e\u003cmrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msubsup\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sum \\limits_{i=1}^{n} x_i\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.3291em;vertical-align:-0.9777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop op-limits\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3514em;\"\u003e\u003cspan style=\"top:-2.1223em;margin-left:0em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003cspan class=\"mrel mtight\"\u003e=\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan\u003e\u003cspan class=\"mop op-symbol small-op\"\u003e∑\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.95em;margin-left:0em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal 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mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\prod_{i=1}^{n} x_i\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmo\u003e∏\u003c/mo\u003e\u003cmrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msubsup\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\prod_{i=1}^{n} x_i\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.104em;vertical-align:-0.2997em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003e\u003cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\"\u003e∏\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8043em;\"\u003e\u003cspan style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003cspan class=\"mrel mtight\"\u003e=\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2997em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\prod\\limits_{i=1}^{n} x_i\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsubsup\u003e\u003cmo\u003e∏\u003c/mo\u003e\u003cmrow\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msubsup\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\prod\\limits_{i=1}^{n} x_i\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.3291em;vertical-align:-0.9777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop op-limits\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3514em;\"\u003e\u003cspan style=\"top:-2.1223em;margin-left:0em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003cspan class=\"mrel mtight\"\u003e=\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan\u003e\u003cspan class=\"mop op-symbol small-op\"\u003e∏\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.95em;margin-left:0em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9777em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"trigonometry\" class=\"headerLink\"\u003e\n \u003ca href=\"#trigonometry\" class=\"header-mark\" aria-label=\"Direct link to section: Trigonometry\"\u003e\u003c/a\u003eTrigonometry\u003c/h3\u003e\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eLaTeX Code\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eResult\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n 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mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\cos(x)\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\cos(x)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003ecos\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\tan(x)\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003etan\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\tan(x)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003etan\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e\u003ccode\u003e\\alpha^\\circ\u003c/code\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eα\u003c/mi\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\alpha^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\"\u003eα\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch3 id=\"matrices\" class=\"headerLink\"\u003e\n \u003ca href=\"#matrices\" class=\"header-mark\" aria-label=\"Direct link to section: Matrices\"\u003e\u003c/a\u003eMatrices\u003c/h3\u003e\u003cp\u003eExample:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 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class=\"cl\"\u003ec \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e d\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ebmatrix\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e[\u003c/mo\u003e\u003cmtable rowspacing=\"0.16em\" columnalign=\"center center\" columnspacing=\"1em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"false\"\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cmo fence=\"true\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{bmatrix}\na \u0026amp; b \\\\\nc \u0026amp; d\n\\end{bmatrix}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e[\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.45em;\"\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.95em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"arraycolsep\" style=\"width:0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"col-align-c\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.45em;\"\u003e\u003cspan style=\"top:-3.61em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.95em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"aligned-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#aligned-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Aligned Equations\"\u003e\u003c/a\u003eAligned Equations\u003c/h3\u003e\u003cp\u003eFor multi-line equations, the \u003ccode\u003ealign\u003c/code\u003e environment is very useful.\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-open show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003elatex\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-3\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\begin\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ealign\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ey \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e= 2x+3 \u003cspan class=\"k\"\u003e\\\\\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003ey-3 \u003cspan class=\"nb\"\u003e\u0026amp;\u003c/span\u003e= 2x\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"k\"\u003e\\end\u003c/span\u003e\u003cspan class=\"nb\"\u003e{\u003c/span\u003ealign\u003cspan class=\"nb\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eResult:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd class =\"mtr-glue\"\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd class =\"mtr-glue\"\u003e\u003c/mtd\u003e\u003cmtd class =\"mml-eqn-num\"\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd class =\"mtr-glue\"\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd class =\"mtr-glue\"\u003e\u003c/mtd\u003e\u003cmtd class =\"mml-eqn-num\"\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{align}\ny \u0026amp;= 2x+3 \\\\\ny-3 \u0026amp;= 2x\n\\end{align}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan 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style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"rendering-with-mathjax\" class=\"headerLink\"\u003e\n \u003ca href=\"#rendering-with-mathjax\" class=\"header-mark\" aria-label=\"Direct link to section: Rendering with MathJax\"\u003e\u003c/a\u003eRendering with MathJax\u003c/h2\u003e\u003cp\u003eIf your blog does not yet have a math renderer, MathJax is one of the most common choices. 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0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-4\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e \u003cspan class=\"na\"\u003etype\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;text/x-mathjax-config\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003eMathJax\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"nx\"\u003eHub\u003c/span\u003e\u003cspan class=\"p\"\u003e.\u003c/span\u003e\u003cspan class=\"nx\"\u003eConfig\u003c/span\u003e\u003cspan class=\"p\"\u003e({\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003etex2jax\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003einlineMath\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;$\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;$\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;\\\\(\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;\\\\)\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e]],\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"nx\"\u003edisplayMath\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"p\"\u003e[[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;$$\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;$$\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e],\u003c/span\u003e \u003cspan class=\"p\"\u003e[\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;\\\\[\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e\u003cspan class=\"s1\"\u003e\u0026#39;\\\\]\u0026#39;\u003c/span\u003e\u003cspan class=\"p\"\u003e]]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e});\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e \u003cspan class=\"na\"\u003esrc\u003c/span\u003e\u003cspan class=\"o\"\u003e=\u003c/span\u003e\u003cspan class=\"s\"\u003e\u0026#34;https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003cp\u003eFor writing formulas:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003einline math can use \u003ccode\u003e$ ... $\u003c/code\u003e or \u003ccode\u003e\\( ... \\)\u003c/code\u003e,\u003c/li\u003e\n\u003cli\u003edisplay math can use \u003ccode\u003e$$ ... $$\u003c/code\u003e or \u003ccode\u003e\\[ ... \\]\u003c/code\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"rendering-with-katex\" class=\"headerLink\"\u003e\n \u003ca href=\"#rendering-with-katex\" class=\"header-mark\" aria-label=\"Direct link to section: Rendering with KaTeX\"\u003e\u003c/a\u003eRendering with KaTeX\u003c/h2\u003e\u003cp\u003eKaTeX is usually lighter and faster. Example setup:\u003c/p\u003e\n\u003cdiv class=\"code-block highlight is-closed show-line-numbers tw-group tw-my-2\"\u003e\n \u003cdiv class=\"\n \n tw-flex \n tw-flex-row\n tw-flex-1 \n tw-justify-between \n tw-w-full tw-bg-bgColor-secondary\n \"\u003e \n \u003cbutton \n class=\"\n code-block-button\n tw-mx-2 \n tw-flex\n tw-flex-row\n tw-flex-1\"\n aria-hidden=\"true\"\u003e\n \u003cdiv class=\"group-[.is-open]:tw-rotate-90 tw-transition-[transform] tw-duration-500 tw-ease-in-out print:!tw-hidden tw-w-min tw-h-min tw-my-1 tw-mx-1\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 320 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M285.476 272.971L91.132 467.314c-9.373 9.373-24.569 9.373-33.941 0l-22.667-22.667c-9.357-9.357-9.375-24.522-.04-33.901L188.505 256 34.484 101.255c-9.335-9.379-9.317-24.544.04-33.901l22.667-22.667c9.373-9.373 24.569-9.373 33.941 0L285.475 239.03c9.373 9.372 9.373 24.568.001 33.941z\"/\u003e\u003c/svg\u003e\u003c/div\u003e\n \u003cp class=\"tw-select-none !tw-my-1\"\u003ehtml\u003c/p\u003e\n \u003c/button\u003e\n\n \u003cdiv class=\"tw-flex\"\u003e\n \u003cbutton \n class=\"\n line-number-button\n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.show-line-numbers]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle line numbers\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M61.77 401l17.5-20.15a19.92 19.92 0 0 0 5.07-14.19v-3.31C84.34 356 80.5 352 73 352H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8h22.83a157.41 157.41 0 0 0-11 12.31l-5.61 7c-4 5.07-5.25 10.13-2.8 14.88l1.05 1.93c3 5.76 6.29 7.88 12.25 7.88h4.73c10.33 0 15.94 2.44 15.94 9.09 0 4.72-4.2 8.22-14.36 8.22a41.54 41.54 0 0 1-15.47-3.12c-6.49-3.88-11.74-3.5-15.6 3.12l-5.59 9.31c-3.72 6.13-3.19 11.72 2.63 15.94 7.71 4.69 20.38 9.44 37 9.44 34.16 0 48.5-22.75 48.5-44.12-.03-14.38-9.12-29.76-28.73-34.88zM496 224H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zm0-160H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16V80a16 16 0 0 0-16-16zm0 320H176a16 16 0 0 0-16 16v32a16 16 0 0 0 16 16h320a16 16 0 0 0 16-16v-32a16 16 0 0 0-16-16zM16 160h64a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H64V40a8 8 0 0 0-8-8H32a8 8 0 0 0-7.14 4.42l-8 16A8 8 0 0 0 24 64h8v64H16a8 8 0 0 0-8 8v16a8 8 0 0 0 8 8zm-3.91 160H80a8 8 0 0 0 8-8v-16a8 8 0 0 0-8-8H41.32c3.29-10.29 48.34-18.68 48.34-56.44 0-29.06-25-39.56-44.47-39.56-21.36 0-33.8 10-40.46 18.75-4.37 5.59-3 10.84 2.8 15.37l8.58 6.88c5.61 4.56 11 2.47 16.12-2.44a13.44 13.44 0 0 1 9.46-3.84c3.33 0 9.28 1.56 9.28 8.75C51 248.19 0 257.31 0 304.59v4C0 316 5.08 320 12.09 320z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n\n \u003cbutton \n class=\"\n wrap-code-button\n tw-select-none \n tw-mx-2 \n tw-hidden \n group-[.is-open]:tw-block \n group-[.is-wrap]:tw-text-fgColor-link \n print:!tw-hidden\" \n title=\"Toggle code wrap\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \n \u003cbutton \n class=\"\n copy-code-button\n tw-select-none\n tw-mx-2 \n tw-hidden\n group-[.is-open]:tw-block\n hover:tw-text-fgColor-link \n print:!tw-hidden\"\n title=\"Copy code\"\u003e\n \u003cspan class=\"copy-icon tw-block\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 448 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M433.941 65.941l-51.882-51.882A48 48 0 0 0 348.118 0H176c-26.51 0-48 21.49-48 48v48H48c-26.51 0-48 21.49-48 48v320c0 26.51 21.49 48 48 48h224c26.51 0 48-21.49 48-48v-48h80c26.51 0 48-21.49 48-48V99.882a48 48 0 0 0-14.059-33.941zM266 464H54a6 6 0 0 1-6-6V150a6 6 0 0 1 6-6h74v224c0 26.51 21.49 48 48 48h96v42a6 6 0 0 1-6 6zm128-96H182a6 6 0 0 1-6-6V54a6 6 0 0 1 6-6h106v88c0 13.255 10.745 24 24 24h88v202a6 6 0 0 1-6 6zm6-256h-64V48h9.632c1.591 0 3.117.632 4.243 1.757l48.368 48.368a6 6 0 0 1 1.757 4.243V112z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003cspan class=\"check-icon tw-hidden\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M173.898 439.404l-166.4-166.4c-9.997-9.997-9.997-26.206 0-36.204l36.203-36.204c9.997-9.998 26.207-9.998 36.204 0L192 312.69 432.095 72.596c9.997-9.997 26.207-9.997 36.204 0l36.203 36.204c9.997 9.997 9.997 26.206 0 36.204l-294.4 294.401c-9.998 9.997-26.207 9.997-36.204-.001z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\n \u003c/button\u003e\n \n \u003cbutton \n class=\"\n tw-select-none \n tw-mx-2 \n tw-block \n group-[.is-open]:tw-hidden \n print:!tw-hidden\" \n disabled\n aria-hidden=\"true\"\u003e\u003csvg class=\"icon\"\n xmlns=\"http://www.w3.org/2000/svg\" viewBox=\"0 0 512 512\"\u003e\u003c!-- Font Awesome Free 5.15.4 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) --\u003e\u003cpath d=\"M328 256c0 39.8-32.2 72-72 72s-72-32.2-72-72 32.2-72 72-72 72 32.2 72 72zm104-72c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72zm-352 0c-39.8 0-72 32.2-72 72s32.2 72 72 72 72-32.2 72-72-32.2-72-72-72z\"/\u003e\u003c/svg\u003e\u003c/button\u003e\n \u003c/div\u003e\n \u003c/div\u003e\n \u003cpre style=\"counter-reset: codeblock;\" class=\"tw-block tw-m-0 tw-p-0\"\u003e\u003ccode \n id=\"codeblock-id-5\" \n class=\"\n chroma \n !tw-block \n tw-p-0\n tw-m-0\n tw-transition-[max-height] \n tw-duration-500 \n tw-ease-in-out \n group-[.is-closed]:!tw-max-h-0 \n group-[.is-wrap]:tw-text-wrap\n tw-overflow-y-hidden\n tw-overflow-x-auto\n tw-scrollbar-thin\n \"\u003e\u003cspan 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class=\"s2\"\u003e\u0026#34;$$\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003edisplay\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"kc\"\u003etrue\u003c/span\u003e\u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\u003cspan class=\"nx\"\u003eleft\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;$\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003eright\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;$\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003edisplay\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"kc\"\u003efalse\u003c/span\u003e\u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\u003cspan class=\"nx\"\u003eleft\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;\\\\(\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003eright\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;\\\\)\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003edisplay\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"kc\"\u003efalse\u003c/span\u003e\u003cspan class=\"p\"\u003e},\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e{\u003c/span\u003e\u003cspan class=\"nx\"\u003eleft\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;\\\\[\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003eright\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"s2\"\u003e\u0026#34;\\\\]\u0026#34;\u003c/span\u003e\u003cspan class=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nx\"\u003edisplay\u003c/span\u003e\u003cspan class=\"o\"\u003e:\u003c/span\u003e \u003cspan class=\"kc\"\u003etrue\u003c/span\u003e\u003cspan class=\"p\"\u003e}\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e]\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e});\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e \u003cspan class=\"p\"\u003e});\u003c/span\u003e\n\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"line\"\u003e\u003cspan class=\"cl\"\u003e\u003cspan class=\"p\"\u003e\u0026lt;/\u003c/span\u003e\u003cspan class=\"nt\"\u003escript\u003c/span\u003e\u003cspan class=\"p\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n\u003ch2 id=\"mathjax-or-katex\" class=\"headerLink\"\u003e\n \u003ca href=\"#mathjax-or-katex\" class=\"header-mark\" aria-label=\"Direct link to section: MathJax or KaTeX?\"\u003e\u003c/a\u003eMathJax or KaTeX?\u003c/h2\u003e\u003cp\u003eIn practice:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eMathJax\u003c/strong\u003e is more flexible and highly compatible,\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eKaTeX\u003c/strong\u003e is faster and lighter,\u003c/li\u003e\n\u003cli\u003efor most educational blogs, either one works well.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"practical-tips\" class=\"headerLink\"\u003e\n \u003ca href=\"#practical-tips\" class=\"header-mark\" aria-label=\"Direct link to section: Practical Tips\"\u003e\u003c/a\u003ePractical Tips\u003c/h2\u003e\u003cp\u003eSome recommendations:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003euse real \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e code instead of screenshots,\u003c/li\u003e\n\u003cli\u003etest formulas in a draft before publishing,\u003c/li\u003e\n\u003cli\u003estay consistent with inline and display delimiters,\u003c/li\u003e\n\u003cli\u003eavoid placing renderer scripts repeatedly inside article content,\u003c/li\u003e\n\u003cli\u003eif your blog template already loads MathJax or KaTeX, the article only needs the formula code itself.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eWriting mathematical formulas on a blog becomes straightforward once the workflow is clear: write the formula in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, paste it into the article, and let MathJax or KaTeX render it. This produces much cleaner results than using formula screenshots and makes mathematics articles easier to read.\u003c/p\u003e\n", "language": "en" }, { "title" : "Sphere", "date_published" : "2021-08-21T14:43:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/01/bola/", "url" : "https://www.aantriono.com/2021/01/bola/", "summary": " IntroductionA sphere is one of the most familiar curved-surface solids in everyday life. Footballs, marbles, and many round containers are good examples. This article covers the elements of a sphere, its surface area, its volume, and how its volume changes when the radius changes.\nA. Basic Competencies3.7 Generalize the surface area and volume of curved-surface solids such as cylinders, cones, and spheres.\n4.7 Solve contextual problems related to the surface area and volume of curved-surface solids and their combinations.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eA sphere is one of the most familiar curved-surface solids in everyday life. Footballs, marbles, and many round containers are good examples. This article covers the elements of a sphere, its surface area, its volume, and how its volume changes when the radius changes.\u003c/p\u003e\n\u003ch2 id=\"a-basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: A. Basic Competencies\"\u003e\u003c/a\u003eA. Basic Competencies\u003c/h2\u003e\u003cp\u003e\u003ccode\u003e3.7\u003c/code\u003e Generalize the surface area and volume of curved-surface solids such as cylinders, cones, and spheres.\u003cbr\u003e\n\u003ccode\u003e4.7\u003c/code\u003e Solve contextual problems related to the surface area and volume of curved-surface solids and their combinations.\u003c/p\u003e\n\u003ch2 id=\"b-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: B. Learning Objectives\"\u003e\u003c/a\u003eB. Learning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying this lesson, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eidentify the elements of a sphere\u003c/li\u003e\n\u003cli\u003ecalculate the surface area of a sphere\u003c/li\u003e\n\u003cli\u003ecalculate the volume of a sphere\u003c/li\u003e\n\u003cli\u003ecompare sphere volumes after a change in radius\u003c/li\u003e\n\u003cli\u003edetermine the change in sphere volume when the radius changes\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-lesson-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-lesson-material\" class=\"header-mark\" aria-label=\"Direct link to section: C. Lesson Material\"\u003e\u003c/a\u003eC. Lesson Material\u003c/h2\u003e\u003ch3 id=\"1-elements-of-a-sphere\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-elements-of-a-sphere\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Elements of a Sphere\"\u003e\u003c/a\u003e1. Elements of a Sphere\u003c/h3\u003e\u003cp\u003eA sphere is a curved solid bounded by a single curved surface. It can be formed by rotating a semicircle through \u003ccode\u003e360°\u003c/code\u003e about its diameter.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-Fp1ILHOLp-fhuRG8p6fXLuBWn5li-jWuAnSPCDY1aqtR8wR5DH3eFdJDFGoDfGCpVI3VaSfF_axB8wuEGC_F929YBxttcL9SgHkX7Er87SoRRd_23Y8LWzT3bf-sTP7uW2FWs0E9pEc/s512/bola1.png' alt=\"Curved-surface solid: sphere\" \u003e\u003c/p\u003e\n\u003cp\u003eImportant elements:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ea sphere has one curved surface\u003c/li\u003e\n\u003cli\u003ethe distance from the center to any point on the surface is the \u003cstrong\u003eradius\u003c/strong\u003e, denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-surface-area-of-a-sphere\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-surface-area-of-a-sphere\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Surface Area of a Sphere\"\u003e\u003c/a\u003e2. Surface Area of a Sphere\u003c/h3\u003e\u003cp\u003eOne way to understand the formula is to compare half of a sphere with a rectangle covered by thread.\u003c/p\u003e\n\u003cp\u003eIllustrations:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVqFi0sKyPYXvZVpihWNulS_DtbIG-H0JHv5EVzYuXNxVKcRea2XaCXXduQwqf4H_Zq_IY6VIqp2C6cj4MAbp7eC5Jwrfb7Vcsz4BsymuKmM7aMJzimQQosKY8Pl07iI7EyoN2-ZlpVqA/s575/bola2.png' alt=\"Measuring sphere surface area\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5r3oDojK4NoEMlfuie01j2IdpFiZr4xcjtfI18ftSZ8e0nSBFSgaYXzdAE6xvzwMmGaR-ZIdEk9yDOH-8-sHhkPtvM05olVO-h-v88dtMOWMrjjuyfpZjSyz3e1bj-mS8ve7jkAynXkA/s273/bola9.png' alt=\"Comparison rectangle\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3ThkWxyk1ARzTpzXOxPu1zQ3Ix-QfFznYgrNDA_Jtb9JbHVMT1ehwPPlw62JAGBz8XAXK5_7HYJ5vRxY3VBatbwjbuJvVSq-IvYuO4W7Kg3NOQL1OjCuuFmkhK7btWXsq8cYb3Xe5RTA/s489/bola4.png' alt=\"Relation to a hemisphere\" \u003e\u003c/p\u003e\n\u003cp\u003eFrom this activity, the surface area of a hemisphere equals the area of a rectangle with:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003elength = circumference of the circle = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\pi r\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003ewidth = radius = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eSurface area of a hemisphere\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003el\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Surface area of a hemisphere}\n\u0026amp;= l \\times w\\\\\n\u0026amp;= 2\\pi r \\times r\\\\\n\u0026amp;= 2\\pi r^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5241em;vertical-align:-2.0121em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eSurface area of a hemisphere\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.01968em;\"\u003el\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02691em;\"\u003ew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the surface area of the whole sphere is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eSurface area of a sphere\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003esurface area of a hemisphere\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Surface area of a sphere}\n\u0026amp;= 2 \\times \\text{surface area of a hemisphere}\\\\\n\u0026amp;= 2 \\times 2\\pi r^2\\\\\n\u0026amp;= 4\\pi r^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eSurface area of a sphere\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003esurface area of a hemisphere\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThus:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = 4\\pi r^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h4\u003e\u003cp\u003eA sphere has radius \u003ccode\u003e7 cm\u003c/code\u003e. Find its surface area.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6vRuYvXow-zmpU4WJJZDVuneWrigiKZVBOXQABgwnMR-wOU1iHtr195VmadByHNT5G4UodizK2nyHI-w07R4z_alHixVgr0c5nku7QERxAU9lNZG8eA65JPbNSlmwA6xa91e5iMKJKAI/s261/bola5.png' alt=\"Sphere illustration\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e616\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nS \u0026amp;= 4\\pi r^2\\\\\n \u0026amp;= 4 \\times \\frac{22}{7} \\times 7^2\\\\\n \u0026amp;= 616\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3315em;vertical-align:-2.4158em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9158em;\"\u003e\u003cspan style=\"top:-5.3731em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3917em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4158em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9158em;\"\u003e\u003cspan style=\"top:-5.3731em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3917em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e616\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4158em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e616\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n616\\ \\text{cm}^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e616\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h4\u003e\u003cp\u003eIf the surface area of a sphere is \u003ccode\u003e154 cm²\u003c/code\u003e, find its radius.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e154\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e154\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e88\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12.25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e12.25\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n154 \u0026amp;= 4 \\times \\frac{22}{7} \\times r^2\\\\\nr^2 \u0026amp;= \\frac{154 \\times 7}{88}\\\\\nr^2 \u0026amp;= 12.25\\\\\nr \u0026amp;= \\sqrt{12.25}\\\\\nr \u0026amp;= 3.5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:9.2551em;vertical-align:-4.3775em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.8775em;\"\u003e\u003cspan style=\"top:-6.8775em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e154\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5701em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.72em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.1039em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.3961em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.3775em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.8775em;\"\u003e\u003cspan style=\"top:-6.8775em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5701em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e88\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e154\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" 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class=\"mord\"\u003e12.25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.1039em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e12.25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.3961em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.3775em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the radius is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3.5\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3.5\\ \\text{cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.5\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example 3\"\u003e\u003c/a\u003eExample 3\u003c/h4\u003e\u003cp\u003eA solid hemisphere has radius \u003ccode\u003e10 cm\u003c/code\u003e. Find its surface area.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjea_ihLAskGQDo9kIZc1Y7RFHl08s83qJYSXF_bdojU-Ad_FPpzHQDZEjYqFKwmR2_ULOH0-mlWllTmN_1nkGw37UG-q9iDo9n-IRIiNXsno7yz4WBVcNTfj3updxv4yTgLIvtnzbK7c4/s211/bola10.png' alt=\"Solid hemisphere\" \u003e\u003c/p\u003e\n\u003cp\u003eSurface area of a solid hemisphere = area of the curved half-sphere + area of the circular base.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e942\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nS \u0026amp;= \\frac{1}{2}(4\\pi r^2)+\\pi r^2\\\\\n \u0026amp;= 2\\pi r^2+\\pi r^2\\\\\n \u0026amp;= 3\\pi r^2\\\\\n \u0026amp;= 3 \\times 3.14 \\times 10^2\\\\\n \u0026amp;= 942\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:8.3798em;vertical-align:-3.9399em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4399em;\"\u003e\u003cspan style=\"top:-6.4399em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5898em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5416em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.0416em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.9399em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4399em;\"\u003e\u003cspan style=\"top:-6.4399em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5898em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5416em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.0416em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e942\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.9399em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e942\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n942\\ \\text{cm}^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e942\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-volume-of-a-sphere\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-volume-of-a-sphere\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Volume of a Sphere\"\u003e\u003c/a\u003e3. Volume of a Sphere\u003c/h3\u003e\u003cp\u003eThe sphere volume formula can be derived by comparing a hemisphere with two cones of radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh12teIPPgHAHeH0jYCrJfZi-YqQwHIRPFSDryy0ftwLsMfTj7xOeYXJ0igBmNTZvgYirSGYXVutyublqCOzbffyPKVfVBQJSuy3n2-Ts3anBrhyphenhyphenLhYCls0hXaGoV34jFB_NHs9iFr_jQ8/s634/bola7.png' alt=\"Sphere and cones\" \u003e\u003c/p\u003e\n\u003cp\u003eFrom that comparison:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eVolume of a hemisphere\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003evolume of a cone\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Volume of a hemisphere}\n\u0026amp;= 2 \\times \\text{volume of a cone}\\\\\n\u0026amp;= 2 \\times \\frac{1}{3}\\pi r^2 t\\\\\n\u0026amp;= 2 \\times \\frac{1}{3}\\pi r^2 r\\\\\n\u0026amp;= \\frac{2}{3}\\pi r^3\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:8.4223em;vertical-align:-3.9612em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4612em;\"\u003e\u003cspan style=\"top:-6.9426em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eVolume of a hemisphere\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.9612em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.3463em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.9612em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4612em;\"\u003e\u003cspan style=\"top:-6.9426em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003evolume of a cone\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.9612em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.3463em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.9612em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eHence the volume of the whole sphere is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eVolume of a sphere\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003evolume of a hemisphere\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Volume of a sphere}\n\u0026amp;= 2 \\times \\text{volume of a hemisphere}\\\\\n\u0026amp;= 2 \\times \\frac{2}{3}\\pi r^3\\\\\n\u0026amp;= \\frac{4}{3}\\pi r^3\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.1149em;vertical-align:-2.8074em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.7889em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eVolume of a sphere\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.7889em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003evolume of a hemisphere\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er=\\frac{1}{2}d\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, we also get:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eVolume of a sphere\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Volume of a sphere}\n\u0026amp;= \\frac{4}{3}\\pi \\left(\\frac{1}{2}d\\right)^3\\\\\n\u0026amp;= \\frac{1}{6}\\pi d^3\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.2115em;vertical-align:-2.3557em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8557em;\"\u003e\u003cspan style=\"top:-4.8557em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eVolume of a sphere\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.2843em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.3557em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8557em;\"\u003e\u003cspan style=\"top:-4.8557em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.2843em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.3557em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV = \\frac{4}{3}\\pi r^3 \\quad \\text{or} \\quad V = \\frac{1}{6}\\pi d^3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-1-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h4\u003e\u003cp\u003eA sphere has radius \u003ccode\u003e21 cm\u003c/code\u003e. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi=\\frac{22}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, find its volume.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eV\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e38,808\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nV \u0026amp;= \\frac{4}{3}\\pi r^3\\\\\n \u0026amp;= \\frac{4}{3}\\times \\frac{22}{7}\\times 21^3\\\\\n \u0026amp;= 38{,}808\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.1149em;vertical-align:-2.8074em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.3074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.3074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e38\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e808\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e38,808\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n38{,}808\\ \\text{cm}^3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0585em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e38\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e808\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-2-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h4\u003e\u003cp\u003eA sphere of radius \u003ccode\u003e6 cm\u003c/code\u003e is placed into a cylinder filled with water so that the water level rises. If the cylinder base radius is \u003ccode\u003e15 cm\u003c/code\u003e, find the rise in water level.\u003c/p\u003e\n\u003cp\u003eLet the sphere radius be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er_a=6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and the cylinder radius be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er_b=15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm. Since the displaced water volume equals the sphere volume:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsubsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msubsup\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsubsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msubsup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\pi r_b^2 t = \\frac{4}{3}\\pi r_a^3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1111em;vertical-align:-0.247em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.247em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.247em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e225\u003c/mn\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e288\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e288\u003c/mn\u003e\u003cmn\u003e225\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1.28\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n15^2 t \u0026amp;= \\frac{4}{3}\\times 6^3\\\\\n225t \u0026amp;= 288\\\\\nt \u0026amp;= \\frac{288}{225}\\\\\nt \u0026amp;= 1.28\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.6149em;vertical-align:-3.5574em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0574em;\"\u003e\u003cspan style=\"top:-6.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.2314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e225\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.424em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0574em;\"\u003e\u003cspan style=\"top:-6.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.2314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e288\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e225\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e288\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.424em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1.28\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the water rises by:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1.28\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n1.28\\ \\text{cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1.28\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"4-comparing-sphere-volumes\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-comparing-sphere-volumes\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Comparing Sphere Volumes\"\u003e\u003c/a\u003e4. Comparing Sphere Volumes\u003c/h3\u003e\u003cp\u003eBecause sphere volume depends on \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, a change in radius has a strong effect on the volume.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdH9UmfMLAb-Xw7pPIzGhanPPuwsRbYfwPK2xARKyg4cGKKFgGf6glV29UbzORbwgmAtBQOF7kqisMtKZhCzdH09AIzp301aAbr6FVkx7KcQ7t4ICmM0FaAMGWKtUIN4qETNCcXqPhNMU/s467/bola8.png' alt=\"Two spheres\" \u003e\u003c/p\u003e\n\u003cp\u003eIf a sphere originally has radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and volume \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eV\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eif the new radius is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er+a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n V:V_{\\text{new}} = r^3:(r+a)^3\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eif the radius becomes \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e times the original radius, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n V:V_{\\text{new}} = 1:k^3\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eA sphere has radius \u003ccode\u003e2 cm\u003c/code\u003e. Find the ratio of the original volume to the new volume in each case:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ethe radius increases by \u003ccode\u003e3 cm\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003ethe radius decreases by \u003ccode\u003e1 cm\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003ethe radius becomes \u003ccode\u003e2\u003c/code\u003e times the original radius\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eSolutions:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eRadius increases by \u003ccode\u003e3 cm\u003c/code\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e125\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n V:V_{\\text{new}}\n \u0026amp;= 2^3:(2+3)^3\\\\\n \u0026amp;= 8:125\n \\end{aligned}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0241em;vertical-align:-1.2621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e125\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eRadius decreases by \u003ccode\u003e1 cm\u003c/code\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n V:V_{\\text{new}}\n \u0026amp;= 2^3:(2-1)^3\\\\\n \u0026amp;= 8:1\n \\end{aligned}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0241em;vertical-align:-1.2621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eRadius becomes \u003ccode\u003e2\u003c/code\u003e times the original:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n V:V_{\\text{new}}\n \u0026amp;= 1:2^3\\\\\n \u0026amp;= 1:8\n \\end{aligned}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0241em;vertical-align:-1.2621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"5-geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: 5. GeoGebra Simulation\"\u003e\u003c/a\u003e5. GeoGebra Simulation\u003c/h2\u003e\u003cp\u003eMove the slider to the right or left to change the sphere radius. Use the \u003ccode\u003eON\u003c/code\u003e and \u003ccode\u003eOFF\u003c/code\u003e buttons to start or stop the surface-area and volume animation.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/pfts6vu9/width/600/height/400/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Bola\" width=\"600px\"\u003e \u003c/iframe\u003e\n\u003ch2 id=\"6-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Evaluation\"\u003e\u003c/a\u003e6. Evaluation\u003c/h2\u003e\u003cp\u003eThere are two jars for storing sugar as shown below.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPfu06jVyQT-9vzfMyiPWUjl-Ao2KTDrj5Bn1TabFIZAwkbKEFaThk0_Ue6qORv9oS4qufMRjEOvGDnBqe09cWmuUU60MOQW8UEjDfWoEEZPM9WeE3A1Wg_dm7QEX0G149WpwRrtOy9AQ/s466/bola11.png' alt=\"Hemisphere and cone jars\" \u003e\u003c/p\u003e\n\u003cp\u003eDetermine which jar can hold more sugar, assuming both jars have the same wall thickness.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThe sphere is an important curved-surface solid because many applied geometry problems depend on its surface area, volume, and radius relationships. Once these formulas are understood, comparison and volume-change problems become much easier.\u003c/p\u003e\n", "language": "en" }, { "title" : "Discriminant Function", "date_published" : "2021-08-16T04:52:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/08/fungsi-diskriminan/", "url" : "https://www.aantriono.com/2021/08/fungsi-diskriminan/", "summary": " IntroductionIn a quadratic equation, there is an important component called the discriminant. The discriminant is used to determine the type of roots of a quadratic equation or quadratic function.\nIt is denoted by DDD and defined as:\nD=b2−4ac D=b^2-4ac D=b2−4acwhere aaa, bbb, and ccc are the real constants in the quadratic equation:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn a quadratic equation, there is an important component called the \u003cstrong\u003ediscriminant\u003c/strong\u003e. The discriminant is used to determine the type of roots of a quadratic equation or quadratic function.\u003c/p\u003e\n\u003cp\u003eIt is denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and defined as:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nD=b^2-4ac\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the real constants in the quadratic equation:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nax^2+bx+c=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"definition-of-the-discriminant\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-of-the-discriminant\" class=\"header-mark\" aria-label=\"Direct link to section: Definition of the Discriminant\"\u003e\u003c/a\u003eDefinition of the Discriminant\u003c/h2\u003e\u003cp\u003eIn general, a discriminant is a quantity that depends on the coefficients of a polynomial and can be used to determine properties of its roots.\u003c/p\u003e\n\u003cp\u003eFor quadratic equations, the discriminant is a practical tool for identifying the nature of the roots without fully solving the equation.\u003c/p\u003e\n\u003ch2 id=\"types-of-roots-of-quadratic-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-roots-of-quadratic-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Types of Roots of Quadratic Equations\"\u003e\u003c/a\u003eTypes of Roots of Quadratic Equations\u003c/h2\u003e\u003cp\u003eThe value of the discriminant determines the type and number of roots of a quadratic equation.\u003c/p\u003e\n\u003ch3 id=\"1-if\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-if\" class=\"header-mark\" aria-label=\"Direct link to section: 1. If D\u0026gt;0D\u0026gt;0D\u0026gt;0\"\u003e\u003c/a\u003e1. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u0026gt;0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h3\u003e\u003cp\u003eThe quadratic equation has two distinct real roots.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a \u003cstrong\u003eperfect square\u003c/strong\u003e, both roots are rational.\u003c/li\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cstrong\u003enot\u003c/strong\u003e a perfect square, both roots are irrational.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-if\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-if\" class=\"header-mark\" aria-label=\"Direct link to section: 2. If D=0D=0D=0\"\u003e\u003c/a\u003e2. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h3\u003e\u003cp\u003eThe quadratic equation has two equal real roots.\u003c/p\u003e\n\u003ch3 id=\"3-if\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-if\" class=\"header-mark\" aria-label=\"Direct link to section: 3. If D\u0026lt;0D\u0026lt;0D\u0026lt;0\"\u003e\u003c/a\u003e3. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u0026lt;0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h3\u003e\u003cp\u003eThe quadratic equation has no real roots. Its roots are imaginary.\u003c/p\u003e\n\u003ch2 id=\"example-problems-and-discussion\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problems-and-discussion\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problems and Discussion\"\u003e\u003c/a\u003eExample Problems and Discussion\u003c/h2\u003e\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eDetermine the type of roots of the following quadratic equations without solving them completely:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2+x-3=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e9x^2+12x+4=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2+2x+3=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003ch5 id=\"a\" class=\"headerLink\"\u003e\n \u003ca href=\"#a\" class=\"header-mark\" aria-label=\"Direct link to section: a. 2x2\u0026#43;x−3=02x^2\u0026#43;x-3=02x2\u0026#43;x−3=0\"\u003e\u003c/a\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2+x-3=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h5\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=-3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nD \u0026amp;= b^2-4ac \\\\\n \u0026amp;= (1)^2-4(2)(-3) \\\\\n \u0026amp;= 1+24 \\\\\n \u0026amp;= 25\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0482em;vertical-align:-2.7741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan 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style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD=25=5^2\u0026gt;0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8532em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the discriminant is a perfect square.\u003c/p\u003e\n\u003cp\u003eSo, the roots are \u003cstrong\u003ereal, distinct, and rational\u003c/strong\u003e.\u003c/p\u003e\n\u003ch5 id=\"b\" class=\"headerLink\"\u003e\n \u003ca href=\"#b\" class=\"header-mark\" aria-label=\"Direct link to section: b. 9x2\u0026#43;12x\u0026#43;4=09x^2\u0026#43;12x\u0026#43;4=09x2\u0026#43;12x\u0026#43;4=0\"\u003e\u003c/a\u003eb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e9x^2+12x+4=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h5\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e144\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e144\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nD \u0026amp;= b^2-4ac \\\\\n \u0026amp;= (12)^2-4(9)(4) \\\\\n \u0026amp;= 144-144 \\\\\n \u0026amp;= 0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0482em;vertical-align:-2.7741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e144\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e144\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the equation has two equal real roots.\u003c/p\u003e\n\u003cp\u003eSo, the roots are \u003cstrong\u003ereal, equal, and rational\u003c/strong\u003e.\u003c/p\u003e\n\u003ch5 id=\"c\" class=\"headerLink\"\u003e\n \u003ca href=\"#c\" class=\"header-mark\" aria-label=\"Direct link to section: c. x2\u0026#43;2x\u0026#43;3=0x^2\u0026#43;2x\u0026#43;3=0x2\u0026#43;2x\u0026#43;3=0\"\u003e\u003c/a\u003ec. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2+2x+3=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h5\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nD \u0026amp;= b^2-4ac \\\\\n \u0026amp;= (2)^2-4(1)(3) \\\\\n \u0026amp;= 4-12 \\\\\n \u0026amp;= -8\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0482em;vertical-align:-2.7741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u0026lt;0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the equation has no real roots.\u003c/p\u003e\n\u003cp\u003eSo, the roots are \u003cstrong\u003eimaginary\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eGiven the quadratic equation:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2+px+25=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eWhat value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e makes the equation have equal real roots?\u003c/p\u003e\n\u003ch4 id=\"answer-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-1\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003cp\u003eFor equal real roots, the discriminant must satisfy:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nD=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThus:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nb^2-4ac \u0026amp;= 0 \\\\\np^2-4(1)(25) \u0026amp;= 0 \\\\\np^2-100 \u0026amp;= 0 \\\\\np^2 \u0026amp;= 100 \\\\\np \u0026amp;= \\pm 10\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.5964em;vertical-align:-3.5482em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0482em;\"\u003e\u003cspan style=\"top:-6.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6118em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.1118em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5482em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0482em;\"\u003e\u003cspan style=\"top:-6.1841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6118em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.1118em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e±\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5482em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\np=\\pm 10\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e±\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThe expression \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb^2-4ac\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called the discriminant of the quadratic equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eax^2+bx+c=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and is denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. This value is what distinguishes the type of roots of the equation.\u003c/p\u003e\n\u003cp\u003eSo, the main use of the discriminant is to determine the type of roots of a quadratic equation quickly.\u003c/p\u003e\n", "language": "en" }, { "title" : "Definition of Sets", "date_published" : "2021-08-13T02:02:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pengertian-himpunan/", "url" : "https://www.aantriono.com/en/pengertian-himpunan/", "summary": " Introduction\nWhen we shop in a fruit market or supermarket, we see fruits grouped by type, such as grapes, mangoes, and oranges. In mathematics, a collection like this can be called a set if its members are clearly defined.\nNot every collection qualifies as a set. The objects in the collection must be identifiable without ambiguity.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgt-g72FMZLmmHNf8WQykQE4xhdroVr7Qtlr-RACsIv-Av7Qjj_xSsZbdJjgW_Eh55XdUAn57gU0GWiCXy1q4x1PlAcEQ7WIcPwF0PGUxXJzRC3N1yfwmDJaKiMQ_g5qv4ILSIlqW6Hc00/s445/Screenshot+2021-08-12+14.57.34.png' alt=\"Fruit supermarket\" \u003e\u003c/p\u003e\n\u003cp\u003eWhen we shop in a fruit market or supermarket, we see fruits grouped by type, such as grapes, mangoes, and oranges. In mathematics, a collection like this can be called a \u003cstrong\u003eset\u003c/strong\u003e if its members are clearly defined.\u003c/p\u003e\n\u003cp\u003eNot every collection qualifies as a set. The objects in the collection must be identifiable without ambiguity.\u003c/p\u003e\n\u003ch2 id=\"definition-of-a-set\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-of-a-set\" class=\"header-mark\" aria-label=\"Direct link to section: Definition of a Set\"\u003e\u003c/a\u003eDefinition of a Set\u003c/h2\u003e\u003cp\u003eExamples of collections that are sets:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe set of male students\u003c/li\u003e\n\u003cli\u003ethe set of four-legged animals\u003c/li\u003e\n\u003cli\u003ethe set of fruits whose names begin with the letter \u003ccode\u003eS\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExamples of collections that are not sets:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe set of school subjects students like\u003c/li\u003e\n\u003cli\u003ethe set of delicious foods\u003c/li\u003e\n\u003cli\u003ethe set of clever students\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eFrom these examples, we can state:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eA set is a collection of objects that is clearly defined.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe objects inside a set are called its \u003cstrong\u003emembers\u003c/strong\u003e or \u003cstrong\u003eelements\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"symbols-and-notation\" class=\"headerLink\"\u003e\n \u003ca href=\"#symbols-and-notation\" class=\"header-mark\" aria-label=\"Direct link to section: Symbols and Notation\"\u003e\u003c/a\u003eSymbols and Notation\u003c/h2\u003e\u003cp\u003eGeneral rules for writing sets:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eset names usually use capital letters such as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003ethe members are written inside curly braces \u003ccode\u003e{ }\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003emembers are separated by commas\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003enatural numbers less than \u003c/mtext\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003eGrade VII students who wear glasses\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nA \u0026amp;= \\{\\text{natural numbers less than } 10\\}\\\\\nB \u0026amp;= \\{2,3,5,7,11\\}\\\\\nC \u0026amp;= \\{\\text{Grade VII students who wear glasses}\\}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003enatural numbers less than \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eGrade VII students who wear glasses\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eCommon number sets include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003enatural numbers\u003c/li\u003e\n\u003cli\u003ewhole numbers\u003c/li\u003e\n\u003cli\u003eintegers\u003c/li\u003e\n\u003cli\u003eeven numbers\u003c/li\u003e\n\u003cli\u003eodd numbers\u003c/li\u003e\n\u003cli\u003eprime numbers\u003c/li\u003e\n\u003cli\u003erational numbers\u003c/li\u003e\n\u003cli\u003eirrational numbers\u003c/li\u003e\n\u003cli\u003ereal numbers\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"ways-to-describe-a-set\" class=\"headerLink\"\u003e\n \u003ca href=\"#ways-to-describe-a-set\" class=\"header-mark\" aria-label=\"Direct link to section: Ways to Describe a Set\"\u003e\u003c/a\u003eWays to Describe a Set\u003c/h2\u003e\u003cp\u003eA set can be written in several ways.\u003c/p\u003e\n\u003ch3 id=\"1-using-words\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-using-words\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Using words\"\u003e\u003c/a\u003e1. Using words\u003c/h3\u003e\u003cp\u003eExample:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003etwo-legged animals\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = \\{\\text{two-legged animals}\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003etwo-legged animals\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-using-set-builder-notation\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-using-set-builder-notation\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Using set-builder notation\"\u003e\u003c/a\u003e2. Using set-builder notation\u003c/h3\u003e\u003cp\u003eExamples:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∣\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmtext\u003e is a prime number less than \u003c/mtext\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∣\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmtext\u003e is an integer\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nK \u0026amp;= \\{x \\mid x \\text{ is a prime number less than } 15\\}\\\\\nL \u0026amp;= \\{x \\mid 2 \u0026lt; x \u0026lt; 8,\\ x \\text{ is an integer}\\}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∣\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e is a prime number less than \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∣\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e is an integer\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-listing-all-members\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-listing-all-members\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Listing all members\"\u003e\u003c/a\u003e3. Listing all members\u003c/h3\u003e\u003cp\u003eExamples:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003eJanuary, February, May, June, July\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003ered, orange, yellow, green, blue, indigo, violet\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nK \u0026amp;= \\{\\text{January, February, May, June, July}\\}\\\\\nL \u0026amp;= \\{\\text{red, orange, yellow, green, blue, indigo, violet}\\}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eJanuary, February, May, June, July\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ered, orange, yellow, green, blue, indigo, violet\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"members-of-a-set\" class=\"headerLink\"\u003e\n \u003ca href=\"#members-of-a-set\" class=\"header-mark\" aria-label=\"Direct link to section: Members of a Set\"\u003e\u003c/a\u003eMembers of a Set\u003c/h2\u003e\u003cp\u003eSuppose:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = \\{1,2,3,4,5\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen \u003ccode\u003e1\u003c/code\u003e, \u003ccode\u003e2\u003c/code\u003e, \u003ccode\u003e3\u003c/code\u003e, \u003ccode\u003e4\u003c/code\u003e, and \u003ccode\u003e5\u003c/code\u003e are members of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, written as:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n1 \\in A,\\ 2 \\in A,\\ 3 \\in A,\\ 4 \\in A,\\ 5 \\in A\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6835em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe symbol \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\in\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e means \u0026ldquo;is a member of\u0026rdquo;, while \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo mathvariant=\"normal\"\u003e∉\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\notin\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"llap\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e/\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.0556em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e means \u0026ldquo;is not a member of\u0026rdquo;.\u003c/p\u003e\n\u003cp\u003eSince \u003ccode\u003e6\u003c/code\u003e is not in the set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, we write:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo mathvariant=\"normal\"\u003e∉\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n6 \\notin A\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"llap\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e/\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.0556em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"number-of-members\" class=\"headerLink\"\u003e\n \u003ca href=\"#number-of-members\" class=\"header-mark\" aria-label=\"Direct link to section: Number of Members\"\u003e\u003c/a\u003eNumber of Members\u003c/h2\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a set, the number of its members is written as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(A)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. This is called the \u003cstrong\u003ecardinality\u003c/strong\u003e of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ea set with finitely many members is a \u003cstrong\u003efinite set\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003ea set with infinitely many members is an \u003cstrong\u003einfinite set\u003c/strong\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eZ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eZ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the set of integers between \u003ccode\u003e0\u003c/code\u003e and \u003ccode\u003e6\u003c/code\u003e, then\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n Z = \\{1,2,3,4,5\\}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eZ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eso:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n n(Z)=5\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eZ\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eN\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the set of natural numbers, then\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e…\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n N = \\{1,2,3,4,5,\\ldots\\}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eso:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∞\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n n(N)=\\infty\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∞\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"the-empty-set\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-empty-set\" class=\"header-mark\" aria-label=\"Direct link to section: The Empty Set\"\u003e\u003c/a\u003eThe Empty Set\u003c/h2\u003e\u003cp\u003e\u003cstrong\u003eDefinition:\u003c/strong\u003e The empty set is a set with no members. Its symbol is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∅\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\varnothing\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6633em;vertical-align:-0.0817em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord amsrm\"\u003e∅\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or \u003ccode\u003e{ }\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eThe empty set is a subset of every set.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eLet \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eH\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e be the set of Indonesian presidents whose names begin with \u003ccode\u003eP\u003c/code\u003e. Since there are none:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eH\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∅\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n H = \\varnothing\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6633em;vertical-align:-0.0817em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord amsrm\"\u003e∅\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eLet \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eM\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e be the set of months with exactly \u003ccode\u003e20\u003c/code\u003e days. Since no such month exists:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∅\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n M = \\varnothing\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6633em;vertical-align:-0.0817em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord amsrm\"\u003e∅\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"subsets\" class=\"headerLink\"\u003e\n \u003ca href=\"#subsets\" class=\"header-mark\" aria-label=\"Direct link to section: Subsets\"\u003e\u003c/a\u003eSubsets\u003c/h2\u003e\u003cp\u003eConsider:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003epedicab, carriage, car, bus, train, ship, airplane\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003ecar, bus, train, airplane, ship\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmtext\u003epedicab, carriage, ship, train, bus\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nA \u0026amp;= \\{\\text{pedicab, carriage, car, bus, train, ship, airplane}\\}\\\\\nB \u0026amp;= \\{\\text{car, bus, train, airplane, ship}\\}\\\\\nC \u0026amp;= \\{\\text{pedicab, carriage, ship, train, bus}\\}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003epedicab, carriage, car, bus, train, ship, airplane\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecar, bus, train, airplane, ship\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003epedicab, carriage, ship, train, bus\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eEvery member of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is also a member of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, so:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e⊂\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB \\subset A\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⊂\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eBut \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is not a subset of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, so:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e⊄\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB \\not\\subset C\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⊂\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eLikewise:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e⊄\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nC \\not\\subset B\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⊂\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIn general:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a subset of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e if every member of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is also a member of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is not a subset of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e if at least one member of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is not in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"number-of-subsets\" class=\"headerLink\"\u003e\n \u003ca href=\"#number-of-subsets\" class=\"header-mark\" aria-label=\"Direct link to section: Number of Subsets\"\u003e\u003c/a\u003eNumber of Subsets\u003c/h2\u003e\u003cp\u003eObserve this pattern:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eif \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{1\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then it has \u003ccode\u003e2\u003c/code\u003e subsets\u003c/li\u003e\n\u003cli\u003eif \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{1,2\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then it has \u003ccode\u003e4\u003c/code\u003e subsets\u003c/li\u003e\n\u003cli\u003eif \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC=\\{1,2,3\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then it has \u003ccode\u003e8\u003c/code\u003e subsets\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eSo we get the formula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eIf \u003c/mtext\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e then the number of subsets of \u003c/mtext\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmtext\u003e is \u003c/mtext\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\text{If } n(A)=n,\\ \\text{then the number of subsets of } A \\text{ is } 2^n\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9088em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ethen the number of subsets of \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e is \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ee\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP=\\{a,b,c,d,e\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ee\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, so \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(P)=5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Therefore the number of subsets is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e32\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n 2^5 = 32\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e∣\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmtext\u003e is a natural number less than \u003c/mtext\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK=\\{x \\mid x \\text{ is a natural number less than } 7\\} = \\{1,2,3,4,5,6\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∣\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e is a natural number less than \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, so \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(K)=6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Therefore the number of subsets is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e64\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n 2^6 = 64\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e64\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"universal-set\" class=\"headerLink\"\u003e\n \u003ca href=\"#universal-set\" class=\"header-mark\" aria-label=\"Direct link to section: Universal Set\"\u003e\u003c/a\u003eUniversal Set\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9SA-LVsoJKMTWD9rOMv5dStWgohDQ6FPPHWyXs9KO5ml4AVVW-MxZ_kEDnvQK-8xuHNM7ZpukcGiATLWIvp5MMCdtbaDWUYAhFcQW5UFaSfbDvilc9XyomIZLS9qH64TWp8hCQ1mB9Vo/s16000/Screenshot+2021-08-12+14.48.13.png' alt=\"Universal set of transportation\" \u003e\u003c/p\u003e\n\u003cp\u003eIf we are discussing cars, buses, trains, and airplanes, then the overall context is \u003cstrong\u003etransportation\u003c/strong\u003e. In set theory, this context is called the \u003cstrong\u003euniversal set\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eThe universal set is usually denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eU\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eU\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eU\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cp\u003eDetermine a possible universal set for:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = \\{2,4,6,8,10\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ePossible universal sets include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe set of whole numbers\u003c/li\u003e\n\u003cli\u003ethe set of natural numbers\u003c/li\u003e\n\u003cli\u003ethe set of integers\u003c/li\u003e\n\u003cli\u003ethe set of even numbers\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eInteractive exercises for this section are available in the original source.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThis article introduced sets, set notation, the empty set, subsets, and the universal set. These are core ideas that support later topics in set theory and logic.\u003c/p\u003e\n", "language": "en" }, { "title" : "Quadratic Equations", "date_published" : "2021-08-08T15:19:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/persamaan-kuadrat/", "url" : "https://www.aantriono.com/en/persamaan-kuadrat/", "summary": "\nIntroductionConsider the following problem.\nA bathroom wall has a rectangular shape and will be covered with square tiles. The length of the wall is 5 tiles longer than its width. If 300 tiles are needed to cover the wall, determine the length of the wall. Assume the area of one tile is 1 square unit.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsW6jCTsZKq57c15398cthVinbwXgxKb9kz9uhxx4n1xLF49CcG-JcLGCScxKovGi1hohaOP3aYDZ7HAWoyPshKWL4lQy4-IM4AA3TkNZ71Dz8n4JXXvqdoRkDafQGkiHzlm331cHd1r4/w640-h479/pk.jpg' alt=\"Quadratic equations\" \u003e\u003c/p\u003e\n\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eConsider the following problem.\u003c/p\u003e\n\u003cp\u003eA bathroom wall has a rectangular shape and will be covered with square tiles. The length of the wall is 5 tiles longer than its width. If 300 tiles are needed to cover the wall, determine the length of the wall. Assume the area of one tile is 1 square unit.\u003c/p\u003e\n\u003cp\u003eProblems like this can be modeled using a quadratic equation.\u003c/p\u003e\n\u003ch2 id=\"definition-of-a-quadratic-equation\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-of-a-quadratic-equation\" class=\"header-mark\" aria-label=\"Direct link to section: Definition of a Quadratic Equation\"\u003e\u003c/a\u003eDefinition of a Quadratic Equation\u003c/h2\u003e\u003cp\u003eA quadratic equation is an equation containing a variable with highest degree two.\u003c/p\u003e\n\u003cp\u003eThe general form is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nax^2+bx+c=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are real numbers,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a constant,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\neq 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"types-of-quadratic-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-quadratic-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Types of Quadratic Equations\"\u003e\u003c/a\u003eTypes of Quadratic Equations\u003c/h2\u003e\u003cp\u003eObserve the following examples.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2-6x+5=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2-4=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2-9x=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eExplanation:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eThe equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2-6x+5=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e has \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=-6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Since all terms are present and nonzero, it is called a \u003cstrong\u003ecomplete quadratic equation\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eThe equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2-4=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e has \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=-4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Since it has no \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e term, it is called a \u003cstrong\u003epure quadratic equation\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eThe equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2-9x=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e has \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=-9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Since its constant term is zero, it is called an \u003cstrong\u003eincomplete quadratic equation\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eSome equations are not written directly in general form, for example:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2+7x=-10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{x}+\\dfrac{1}{x-1}=\\dfrac{3}{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eSuch equations can still be transformed into the general quadratic form using algebraic operations.\u003c/p\u003e\n\u003ch3 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eRewrite \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x^2=3x+20\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e into general form, then determine \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n2x^2 \u0026amp;= 3x+20 \\\\\n2x^2-3x \u0026amp;= 20 \\\\\n2x^2-3x-20 \u0026amp;= 0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5723em;vertical-align:-2.0362em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=-3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=-20\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"roots-of-quadratic-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#roots-of-quadratic-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Roots of Quadratic Equations\"\u003e\u003c/a\u003eRoots of Quadratic Equations\u003c/h2\u003e\u003cp\u003eA value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e that makes the equation true is called a \u003cstrong\u003eroot\u003c/strong\u003e or \u003cstrong\u003esolution\u003c/strong\u003e of the quadratic equation.\u003c/p\u003e\n\u003cp\u003eIn general, the roots can be found by:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003efactoring,\u003c/li\u003e\n\u003cli\u003ecompleting the square,\u003c/li\u003e\n\u003cli\u003eusing the quadratic formula.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"1-factoring\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-factoring\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Factoring\"\u003e\u003c/a\u003e1. Factoring\u003c/h2\u003e\u003cp\u003eFactoring changes an algebraic expression into a product form.\u003c/p\u003e\n\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eFind the roots of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n4x^2-25=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSolution:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n4x^2-25 \u0026amp;= 0 \\\\\n(2x)^2-5^2 \u0026amp;= 0 \\\\\n(2x-5)(2x+5) \u0026amp;= 0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n2x-5 \u0026amp;= 0 \\quad \\text{or} \\quad 2x+5=0 \\\\\n2x \u0026amp;= 5 \\quad \\text{or} \\quad 2x=-5 \\\\\nx \u0026amp;= \\frac{5}{2} \\quad \\text{or} \\quad x=-\\frac{5}{2}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3074em;vertical-align:-2.4037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThus, the roots are:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=\\frac{5}{2} \\quad \\text{or} \\quad x=-\\frac{5}{2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eFind the solution set of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3x^2-7x=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSolution:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n3x^2-7x \u0026amp;= 0 \\\\\nx(3x-7) \u0026amp;= 0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0241em;vertical-align:-1.2621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eHence:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx \u0026amp;= 0 \\quad \\text{or} \\quad 3x-7=0 \\\\\nx \u0026amp;= 0 \\quad \\text{or} \\quad 3x=7 \\\\\nx \u0026amp;= 0 \\quad \\text{or} \\quad x=\\frac{7}{3}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3074em;vertical-align:-2.4037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the solution set is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left\\{0,\\frac{7}{3}\\right\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example 3\"\u003e\u003c/a\u003eExample 3\u003c/h3\u003e\u003cp\u003eFind the roots of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2-7x+10=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFind two numbers whose sum is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and whose product is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, namely \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx^2-7x+10 \u0026amp;= 0 \\\\\n(x-5)(x-2) \u0026amp;= 0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0241em;vertical-align:-1.2621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThus:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx-5 \u0026amp;= 0 \\quad \\text{or} \\quad x-2=0 \\\\\nx \u0026amp;= 5 \\quad \\text{or} \\quad x=2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the roots are:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=5 \\quad \\text{or} \\quad x=2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"related-reading\" class=\"headerLink\"\u003e\n \u003ca href=\"#related-reading\" class=\"header-mark\" aria-label=\"Direct link to section: Related Reading\"\u003e\u003c/a\u003eRelated Reading\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2020/09/fungsi-kuadrat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eQuadratic Function\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2020/09/penyelesaian-persamaan-kuadrat-dengan/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eSolving Quadratic Equations\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2020/09/penilaian-materi-persamaan-kuadrat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eAssessment of Quadratic Equations\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"2-completing-the-square\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-completing-the-square\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Completing the Square\"\u003e\u003c/a\u003e2. Completing the Square\u003c/h2\u003e\u003cp\u003eNot all quadratic equations are easy to solve by factoring. Another method is \u003cstrong\u003ecompleting the square\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eSteps:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003emake sure the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is 1,\u003c/li\u003e\n\u003cli\u003emove the constant to the right side if needed,\u003c/li\u003e\n\u003cli\u003eadd the square of half the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to both sides,\u003c/li\u003e\n\u003cli\u003erewrite into perfect-square form,\u003c/li\u003e\n\u003cli\u003esolve using square roots.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"example-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-4\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eSolve:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2-6x+8=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSolution:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmtext\u003e  \u003c/mtext\u003e\u003cmo\u003e⟺\u003c/mo\u003e\u003cmtext\u003e  \u003c/mtext\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2-6x+8=0 \\iff x^2-6x=-8\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6684em;vertical-align:-0.024em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⟺\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eHalf of the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{-6}{2}=-3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIts square is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(-3)^2=9\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAdd 9 to both sides:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx^2-6x \u0026amp;= -8 \\\\\nx^2-6x+9 \u0026amp;= -8+9 \\\\\n(x-3)^2 \u0026amp;= 1\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5723em;vertical-align:-2.0362em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx-3 \u0026amp;= \\pm 1 \\\\\nx \u0026amp;= 3 \\pm 1\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e±\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=4 \\quad \\text{or} \\quad x=2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the solution set is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\{2,4\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"3-quadratic-formula\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-quadratic-formula\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Quadratic Formula\"\u003e\u003c/a\u003e3. Quadratic Formula\u003c/h2\u003e\u003cp\u003eBesides factoring and completing the square, a quadratic equation can be solved using the \u003cstrong\u003equadratic formula\u003c/strong\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx_{1,2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7167em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct mtight\"\u003e,\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2764em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5904em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9134em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8734em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1266em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIt is derived as follows:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nax^2+bx+c \u0026amp;= 0 \\\\\nax^2+bx \u0026amp;= -c \\\\\nx^2+\\frac{b}{a}x \u0026amp;= -\\frac{c}{a}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.4057em;vertical-align:-2.4528em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9528em;\"\u003e\u003cspan style=\"top:-5.4602em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.9361em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.9046em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4528em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9528em;\"\u003e\u003cspan style=\"top:-5.4602em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.9361em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.9046em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4528em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAdd the following to both sides:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left(\\frac{1}{2}\\cdot \\frac{b}{a}\\right)^2=\\left(\\frac{b}{2a}\\right)^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.604em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.604em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx^2+\\frac{b}{a}x+\\left(\\frac{b}{2a}\\right)^2 \u0026amp;= -\\frac{c}{a}+\\left(\\frac{b}{2a}\\right)^2 \\\\\n\\left(x+\\frac{b}{2a}\\right)^2 \u0026amp;= -\\frac{c}{a}+\\frac{b^2}{4a^2} \\\\\n\\left(x+\\frac{b}{2a}\\right)^2 \u0026amp;= \\frac{-4ac+b^2}{4a^2} \\\\\n\\left(x+\\frac{b}{2a}\\right)^2 \u0026amp;= \\frac{b^2-4ac}{4a^2} \\\\\nx+\\frac{b}{2a} \u0026amp;= \\pm \\sqrt{\\frac{b^2-4ac}{4a^2}} \\\\\nx \u0026amp;= -\\frac{b}{2a}\\pm \\sqrt{\\frac{b^2-4ac}{4a^2}} \\\\\nx \u0026amp;= \\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:19.6725em;vertical-align:-9.5863em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:10.0863em;\"\u003e\u003cspan style=\"top:-12.1317em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-9.2277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.3236em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4196em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.4701em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:2.2699em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:4.9008em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:9.5863em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:10.0863em;\"\u003e\u003cspan style=\"top:-12.1317em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-9.2277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.3236em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4196em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.4701em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e±\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.6994em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-4.4em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.4em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4171em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 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reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.6594em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.4em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:2.48em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7406em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:2.2699em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.6994em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan 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class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4171em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8734em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1266em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:9.5863em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe quantity \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb^2-4ac\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called the \u003cstrong\u003ediscriminant\u003c/strong\u003e, denoted by \u003cstrong\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-5\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eFind the roots of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx^2-x-6=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eKnown:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=-1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=-6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eUse the quadratic formula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" 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class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5574em;\"\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the roots are:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=3 \\quad \\text{and} \\quad x=-2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eAfter studying this material, complete the evaluation by following these steps:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eclick \u003ccode\u003eStart Quiz\u003c/code\u003e,\u003c/li\u003e\n\u003cli\u003echoose the correct answer,\u003c/li\u003e\n\u003cli\u003eclick \u003ccode\u003eNext\u003c/code\u003e for the next question,\u003c/li\u003e\n\u003cli\u003eclick \u003ccode\u003ePrev\u003c/code\u003e to return to the previous question,\u003c/li\u003e\n\u003cli\u003eclick \u003ccode\u003eReset\u003c/code\u003e to start again,\u003c/li\u003e\n\u003cli\u003eclick \u003ccode\u003eSubmit\u003c/code\u003e after finishing all questions,\u003c/li\u003e\n\u003cli\u003echeck the score displayed at the bottom,\u003c/li\u003e\n\u003cli\u003efinal value equals total score divided by maximum score, then multiplied by 100.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat concludes this material on quadratic equations for Grade IX SMP/MTs semester 1. If you find any mistakes, please leave a correction in the comments.\u003c/p\u003e\n", "language": "en" }, { "title" : "Get to know the Merdeka Mengajar Platform", "date_published" : "2021-08-03T10:48:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mengenal-platform-merdeka-mengajar_3/", "url" : "https://www.aantriono.com/en/mengenal-platform-merdeka-mengajar_3/", "summary": " IntroductionOn Monday, August 2, 2021, the Institute for the Development and Empowerment of School Principals and School Supervisors (LPPKS) held online training on data-based planning and management.\nThe activities were divided into two formats:\nsynchronous, through Google Meet video conferences, asynchronous, through Moodle LMS. For Learning Platform Guidance Class 81, the LMS admin and contact person was Mrs. Alfiatun Dwi Budiasih, S.Pd, while the speaker was Mr. Deny Avianto, S.T from Pusdatin Kemdikbudristekdikti.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eOn Monday, August 2, 2021, the Institute for the Development and Empowerment of School Principals and School Supervisors (LPPKS) held online training on data-based planning and management.\u003c/p\u003e\n\u003cp\u003eThe activities were divided into two formats:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003esynchronous\u003c/strong\u003e, through Google Meet video conferences,\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003easynchronous\u003c/strong\u003e, through Moodle LMS.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eFor Learning Platform Guidance Class 81, the LMS admin and contact person was \u003cstrong\u003eMrs. Alfiatun Dwi Budiasih, S.Pd\u003c/strong\u003e, while the speaker was \u003cstrong\u003eMr. Deny Avianto, S.T\u003c/strong\u003e from Pusdatin Kemdikbudristekdikti.\u003c/p\u003e\n\u003ch2 id=\"training-participants\" class=\"headerLink\"\u003e\n \u003ca href=\"#training-participants\" class=\"header-mark\" aria-label=\"Direct link to section: Training Participants\"\u003e\u003c/a\u003eTraining Participants\u003c/h2\u003e\u003cp\u003eThe training participants were school representatives invited by LPPKS from seven provinces. The participant document can be viewed below.\u003c/p\u003e\n\u003ciframe allow=\"autoplay\" height=\"480\" scrolling=\"yes\" src=\"https://drive.google.com/file/d/1xEKeXkXbXOqSMGb9XXcl79whruzqNRsE/preview\" width=\"640\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"merdeka-mengajar-application\" class=\"headerLink\"\u003e\n \u003ca href=\"#merdeka-mengajar-application\" class=\"header-mark\" aria-label=\"Direct link to section: Merdeka Mengajar Application\"\u003e\u003c/a\u003eMerdeka Mengajar Application\u003c/h2\u003e\u003cp\u003eThis training focused on introducing and using the Merdeka Mengajar Platform in classroom practice. At that time, the platform was still available in beta form.\u003c/p\u003e\n\u003cp\u003eIn general, the Merdeka Mengajar Platform is a technology platform designed to help teachers:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eteach more effectively,\u003c/li\u003e\n\u003cli\u003eimprove their competence,\u003c/li\u003e\n\u003cli\u003egrow professionally,\u003c/li\u003e\n\u003cli\u003eaccess teaching resources and assessment tools.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"teaching-resource-features\" class=\"headerLink\"\u003e\n \u003ca href=\"#teaching-resource-features\" class=\"header-mark\" aria-label=\"Direct link to section: Teaching Resource Features\"\u003e\u003c/a\u003eTeaching Resource Features\u003c/h3\u003e\u003cp\u003eThe teaching resource section helps teachers find learning materials based on subject and learning phase. The materials may include:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eteaching modules,\u003c/li\u003e\n\u003cli\u003etextbooks,\u003c/li\u003e\n\u003cli\u003elearning videos,\u003c/li\u003e\n\u003cli\u003eproject modules,\u003c/li\u003e\n\u003cli\u003eother supporting materials.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eUsers can:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003esearch for resources by subject and phase,\u003c/li\u003e\n\u003cli\u003eread and review them,\u003c/li\u003e\n\u003cli\u003ebookmark them,\u003c/li\u003e\n\u003cli\u003esave them into folders,\u003c/li\u003e\n\u003cli\u003edownload them to a device.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eDifference between \u003cstrong\u003ebookmarking\u003c/strong\u003e and \u003cstrong\u003edownloading\u003c/strong\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003ebookmarking\u003c/strong\u003e keeps the resource easy to access and organize inside the app,\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003edownloading\u003c/strong\u003e stores the file directly on the device.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThese features are available to teachers and principals in schools participating in the \u003cstrong\u003eSekolah Penggerak\u003c/strong\u003e program and the \u003cstrong\u003eVocational School Excellence Program\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"student-assessment-features\" class=\"headerLink\"\u003e\n \u003ca href=\"#student-assessment-features\" class=\"header-mark\" aria-label=\"Direct link to section: Student Assessment Features\"\u003e\u003c/a\u003eStudent Assessment Features\u003c/h3\u003e\u003cp\u003eThe student assessment section helps teachers gather information about student learning processes and outcomes. Its main purpose is to identify student competency levels.\u003c/p\u003e\n\u003cp\u003eKey points:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ediagnostic assessments are available in the assessment section,\u003c/li\u003e\n\u003cli\u003eassessments can be searched by subject and phase,\u003c/li\u003e\n\u003cli\u003ethey can be distributed online or offline,\u003c/li\u003e\n\u003cli\u003ethe results can be linked to suitable teaching resource recommendations.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eWho can access this feature:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eteachers and principals in schools included in the Sekolah Penggerak program,\u003c/li\u003e\n\u003cli\u003evocational schools in the excellence program.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eWhen diagnostic assessments became available:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003esince \u003cstrong\u003eJuly 12, 2021\u003c/strong\u003e for elementary, junior high, and senior high school teachers in the Sekolah Penggerak program, as well as Center of Excellence vocational schools.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eWhen they should be used:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eideally at the beginning of the school year so the results can support later teaching plans.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIf the assessment feature does not appear:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003emake sure the Merdeka Mengajar application has been updated to version \u003ccode\u003e0.1.3\u003c/code\u003e,\u003c/li\u003e\n\u003cli\u003echeck the update information in the Play Store,\u003c/li\u003e\n\u003cli\u003ereinstall the application if the feature still does not appear.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"forms-of-assessment-results\" class=\"headerLink\"\u003e\n \u003ca href=\"#forms-of-assessment-results\" class=\"header-mark\" aria-label=\"Direct link to section: Forms of Assessment Results\"\u003e\u003c/a\u003eForms of Assessment Results\u003c/h3\u003e\u003cp\u003eAssessment results are presented in two forms:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eclass-level results\u003c/strong\u003e, showing the competency level of most students in a class;\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eindividual results\u003c/strong\u003e, showing a more detailed analysis for each student.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"assessment-distribution-and-scoring\" class=\"headerLink\"\u003e\n \u003ca href=\"#assessment-distribution-and-scoring\" class=\"header-mark\" aria-label=\"Direct link to section: Assessment Distribution and Scoring\"\u003e\u003c/a\u003eAssessment Distribution and Scoring\u003c/h3\u003e\u003cp\u003eAssessments can be distributed in two ways:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eonline\u003c/strong\u003e, where scoring can happen automatically inside the platform;\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eoffline\u003c/strong\u003e, where scoring must be done manually using the provided answer key.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"user-manual-for-the-merdeka-mengajar-application\" class=\"headerLink\"\u003e\n \u003ca href=\"#user-manual-for-the-merdeka-mengajar-application\" class=\"header-mark\" aria-label=\"Direct link to section: User Manual for the Merdeka Mengajar Application\"\u003e\u003c/a\u003eUser Manual for the Merdeka Mengajar Application\u003c/h2\u003e\u003cp\u003eThe following guide explains how to download the Merdeka Mengajar application from the Play Store and begin using it. Make sure the application has already been installed on a phone or tablet before following the guide.\u003c/p\u003e\n\u003ciframe allow=\"autoplay\" height=\"480\" src=\"https://drive.google.com/file/d/1lprqSCFGPo9rLGpF984eGkZ8i1lIi0FI/preview\" width=\"650\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThe Merdeka Mengajar Platform was introduced to support the Sekolah Penggerak program and the broader Merdeka Mengajar initiative. Through this platform, teachers gain easier access to teaching resources, assessments, and learning support materials.\u003c/p\u003e\n\u003cp\u003eThanks are due to Mrs. Alfiatun Dwi Budiasih, S.Pd as LPPKS contact person and admin, Mr. Deny Avianto, S.T as speaker from Pusdatin Kemdikbudristekdikti, as well as the Head of SMPN Purworejo and related stakeholders who contributed their knowledge about this platform.\u003c/p\u003e\n\u003cp\u003eReference: \u003ca href=\"https://guru.kemdikbud.go.id\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://guru.kemdikbud.go.id\u003c/a\u003e\u003c/p\u003e\n", "language": "en" }, { "title" : "Easy Ways to Find LCM and GCF", "date_published" : "2021-07-31T02:48:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-mudah-mencari-kpk-dan-fpb/", "url" : "https://www.aantriono.com/en/cara-mudah-mencari-kpk-dan-fpb/", "summary": " IntroductionConsider this example. Naomi buys chicken noodles every 2 days, Uciha every 3 days, and Sakiro every 5 days. If they all buy noodles together today, after how many days will they meet again on the same day?\nThis kind of problem can be solved using the idea of the Least Common Multiple (LCM).\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eConsider this example. Naomi buys chicken noodles every \u003ccode\u003e2\u003c/code\u003e days, Uciha every \u003ccode\u003e3\u003c/code\u003e days, and Sakiro every \u003ccode\u003e5\u003c/code\u003e days. If they all buy noodles together today, after how many days will they meet again on the same day?\u003c/p\u003e\n\u003cp\u003eThis kind of problem can be solved using the idea of the \u003cstrong\u003eLeast Common Multiple (LCM)\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"least-common-multiple\" class=\"headerLink\"\u003e\n \u003ca href=\"#least-common-multiple\" class=\"header-mark\" aria-label=\"Direct link to section: Least Common Multiple\"\u003e\u003c/a\u003eLeast Common Multiple\u003c/h2\u003e\u003cp\u003eThe LCM of two or more numbers can be found in several ways:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003elisting multiples\u003c/li\u003e\n\u003cli\u003eprime factorization\u003c/li\u003e\n\u003cli\u003erepeated division\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"listing-multiples\" class=\"headerLink\"\u003e\n \u003ca href=\"#listing-multiples\" class=\"header-mark\" aria-label=\"Direct link to section: Listing Multiples\"\u003e\u003c/a\u003eListing Multiples\u003c/h3\u003e\u003cp\u003eTo understand the LCM, look at these multiples:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003emultiples of \u003ccode\u003e2\u003c/code\u003e: \u003ccode\u003e2, 4, 6, 8, 10, 12, ...\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003emultiples of \u003ccode\u003e3\u003c/code\u003e: \u003ccode\u003e3, 6, 9, 12, 15, ...\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003emultiples of \u003ccode\u003e5\u003c/code\u003e: \u003ccode\u003e5, 10, 15, 20, 25, ...\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eNumbers that appear in more than one list are called \u003cstrong\u003ecommon multiples\u003c/strong\u003e. The smallest of them is the \u003cstrong\u003eleast common multiple\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eThe LCM of \u003ccode\u003e1\u003c/code\u003e and \u003ccode\u003e3\u003c/code\u003e is \u003ccode\u003e3\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eThe LCM of \u003ccode\u003e2\u003c/code\u003e and \u003ccode\u003e5\u003c/code\u003e is \u003ccode\u003e10\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eThe LCM of \u003ccode\u003e4\u003c/code\u003e and \u003ccode\u003e7\u003c/code\u003e is \u003ccode\u003e28\u003c/code\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"prime-factorization\" class=\"headerLink\"\u003e\n \u003ca href=\"#prime-factorization\" class=\"header-mark\" aria-label=\"Direct link to section: Prime Factorization\"\u003e\u003c/a\u003ePrime Factorization\u003c/h3\u003e\u003cp\u003eAnother way to find the LCM is by prime factorization.\u003c/p\u003e\n\u003cp\u003eExample: find the LCM of \u003ccode\u003e90\u003c/code\u003e and \u003ccode\u003e168\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj52f_fUHvuRUDcPz3oAu_s1ZWSs0NTmdyigknkNnsUMLBcPKovjj6SFk0fc1QxKx5gxij6sQvREb5By6eQTCtUyxNLYvE0wjKVmJzoG3gd7muEEEioefVL9UVUVBOICnYJT8ah357s9Wk/s675/Screenshot+2021-07-30+21.34.36.png' alt=\"Finding the LCM with prime factorization\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e90\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e168\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n90 \u0026amp;= 2 \\times 3^2 \\times 5\\\\\n168 \u0026amp;= 2^3 \\times 3 \\times 7\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0482em;vertical-align:-1.2741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e90\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e168\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTo find the LCM, include every prime factor that appears and choose the highest power of each one.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLCM\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2520\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\text{LCM} = 2^3 \\times 3^2 \\times 5 \\times 7 = 2520\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eLCM\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2520\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"lcm-calculator\" class=\"headerLink\"\u003e\n \u003ca href=\"#lcm-calculator\" class=\"header-mark\" aria-label=\"Direct link to section: LCM Calculator\"\u003e\u003c/a\u003eLCM Calculator\u003c/h3\u003e\u003cp\u003eThe interactive calculator for this section is available in the original source.\u003c/p\u003e\n\u003ch3 id=\"repeated-division\" class=\"headerLink\"\u003e\n \u003ca href=\"#repeated-division\" class=\"header-mark\" aria-label=\"Direct link to section: Repeated Division\"\u003e\u003c/a\u003eRepeated Division\u003c/h3\u003e\u003cp\u003eThe LCM can also be found by repeated division. Divide the numbers step by step until all quotients become \u003ccode\u003e1\u003c/code\u003e, then multiply all divisors used.\u003c/p\u003e\n\u003cp\u003eExample: find the LCM of \u003ccode\u003e9\u003c/code\u003e, \u003ccode\u003e15\u003c/code\u003e, and \u003ccode\u003e42\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLHZICCZJNy7geWy6PP85wW78cr0jkUSOlOFgblUgTD9X3_JjotTc78srVKrq7tuif4AQq4K1oh2AQqt-dudUiqltcLbIhGDfjrDkjpk_xjYz0eR_r5ugSE-lJ-OqIPfaBQ4QZGgcDe1M/s505/Screenshot+2021-07-31+09.53.25.png' alt=\"Finding the LCM with repeated division\" \u003e\u003c/p\u003e\n\u003cp\u003eThe result is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e630\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3 \\times 2 \\times 7 \\times 5 \\times 3 = 630\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e630\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the LCM of \u003ccode\u003e9\u003c/code\u003e, \u003ccode\u003e15\u003c/code\u003e, and \u003ccode\u003e42\u003c/code\u003e is \u003ccode\u003e630\u003c/code\u003e.\u003c/p\u003e\n\u003ch2 id=\"common-factors\" class=\"headerLink\"\u003e\n \u003ca href=\"#common-factors\" class=\"header-mark\" aria-label=\"Direct link to section: Common Factors\"\u003e\u003c/a\u003eCommon Factors\u003c/h2\u003e\u003cp\u003eA number \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called a factor of an integer \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e if \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e divides \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e exactly. In other words:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nb = a \\times n\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003efor some integer \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003efactors of \u003ccode\u003e6\u003c/code\u003e: \u003ccode\u003e1, 2, 3, 6\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003efactors of \u003ccode\u003e8\u003c/code\u003e: \u003ccode\u003e1, 2, 4, 8\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003efactors of \u003ccode\u003e24\u003c/code\u003e: \u003ccode\u003e1, 2, 3, 4, 6, 8, 12, 24\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe common factors of \u003ccode\u003e6\u003c/code\u003e and \u003ccode\u003e8\u003c/code\u003e are \u003ccode\u003e{1, 2}\u003c/code\u003e.\u003c/p\u003e\n\u003ch2 id=\"greatest-common-factor\" class=\"headerLink\"\u003e\n \u003ca href=\"#greatest-common-factor\" class=\"header-mark\" aria-label=\"Direct link to section: Greatest Common Factor\"\u003e\u003c/a\u003eGreatest Common Factor\u003c/h2\u003e\u003cp\u003eThe \u003cstrong\u003eGreatest Common Factor (GCF)\u003c/strong\u003e of two or more numbers is the largest positive integer that divides all of them exactly.\u003c/p\u003e\n\u003cp\u003eThe GCF can be found by:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ecomparing factor lists\u003c/li\u003e\n\u003cli\u003eusing prime factorization\u003c/li\u003e\n\u003cli\u003eusing repeated division\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eFind the GCF of \u003ccode\u003e20\u003c/code\u003e and \u003ccode\u003e12\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eFactors of \u003ccode\u003e20\u003c/code\u003e: \u003ccode\u003e1, 2, 4, 5, 10, 20\u003c/code\u003e\u003cbr\u003e\nFactors of \u003ccode\u003e12\u003c/code\u003e: \u003ccode\u003e1, 2, 3, 4, 6, 12\u003c/code\u003e\u003c/p\u003e\n\u003cp\u003eThe largest common factor is \u003ccode\u003e4\u003c/code\u003e, so the GCF of \u003ccode\u003e20\u003c/code\u003e and \u003ccode\u003e12\u003c/code\u003e is \u003ccode\u003e4\u003c/code\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eFind the GCF of \u003ccode\u003e90\u003c/code\u003e and \u003ccode\u003e168\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEiF6q80T-ZhKq_coX6UP3CbdXlWWpdmDVOanv1eBkL1J0cQhqnM18fpUvZnStCW4f6xv-boMIv1nVbRpMqF18sUzJBHsBUwJUhdXUKL6JbKvmrGI_3kpfUyJaMWvcS1Vh_SpNzUfGLZk/s675/Screenshot+2021-07-30+21.34.36.png' alt=\"Finding the GCF with prime factorization\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e90\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e168\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n90 \u0026amp;= 2 \\times 3^2 \\times 5\\\\\n168 \u0026amp;= 2^3 \\times 3 \\times 7\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0482em;vertical-align:-1.2741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e90\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e168\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTo find the GCF, multiply the common prime factors using the lowest powers:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eGCF\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\text{GCF} = 2 \\times 3 = 6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eGCF\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example 3\"\u003e\u003c/a\u003eExample 3\u003c/h3\u003e\u003cp\u003eFind the GCF of \u003ccode\u003e24\u003c/code\u003e, \u003ccode\u003e48\u003c/code\u003e, and \u003ccode\u003e72\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj81vSPCCsW1Q38tVbFWODuepU2N1qDV1wEkjrWLvESF1cuEZ4w471bDRiXr_EhlhFStb_xQX7xKKqZlO4IvAJLvYyAHlQ6GFayP-YVM70G7yiWvrk1U328lhl_Zodf8-ysTUBLViDKmI/s505/Screenshot+2021-07-31+09.55.32.png' alt=\"Finding the GCF with repeated division\" \u003e\u003c/p\u003e\n\u003cp\u003eMultiply the divisors that divide all three numbers:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2 \\times 2 \\times 3 = 12\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the GCF of \u003ccode\u003e24\u003c/code\u003e, \u003ccode\u003e48\u003c/code\u003e, and \u003ccode\u003e72\u003c/code\u003e is \u003ccode\u003e12\u003c/code\u003e.\u003c/p\u003e\n\u003ch3 id=\"gcf-calculator\" class=\"headerLink\"\u003e\n \u003ca href=\"#gcf-calculator\" class=\"header-mark\" aria-label=\"Direct link to section: GCF Calculator\"\u003e\u003c/a\u003eGCF Calculator\u003c/h3\u003e\u003cp\u003eThe interactive calculator for this section is available in the original source.\u003c/p\u003e\n\u003ch2 id=\"solving-everyday-problems-with-lcm-and-gcf\" class=\"headerLink\"\u003e\n \u003ca href=\"#solving-everyday-problems-with-lcm-and-gcf\" class=\"header-mark\" aria-label=\"Direct link to section: Solving Everyday Problems with LCM and GCF\"\u003e\u003c/a\u003eSolving Everyday Problems with LCM and GCF\u003c/h2\u003e\u003ch3 id=\"1-motorbike-service-schedule\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-motorbike-service-schedule\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Motorbike service schedule\"\u003e\u003c/a\u003e1. Motorbike service schedule\u003c/h3\u003e\u003cp\u003eAni has two motorbikes. One must be serviced every \u003ccode\u003e30\u003c/code\u003e days, the other every \u003ccode\u003e25\u003c/code\u003e days. After how many days will both need service on the same day?\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e30\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n25 \u0026amp;= 5^2\\\\\n30 \u0026amp;= 2 \\times 3 \\times 5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0241em;vertical-align:-1.2621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7621em;\"\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe LCM is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e150\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2 \\times 3 \\times 5^2 = 150\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e150\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo both motorbikes will be serviced together every \u003ccode\u003e150\u003c/code\u003e days.\u003c/p\u003e\n\u003ch3 id=\"2-packing-bread-into-bags\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-packing-bread-into-bags\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Packing bread into bags\"\u003e\u003c/a\u003e2. Packing bread into bags\u003c/h3\u003e\u003cp\u003eAndi has \u003ccode\u003e15\u003c/code\u003e chocolate breads and \u003ccode\u003e20\u003c/code\u003e banana breads. He wants to pack them into bags with the same composition in each bag. How many bags can be made?\u003c/p\u003e\n\u003cp\u003eFactors of \u003ccode\u003e15\u003c/code\u003e: \u003ccode\u003e1, 3, 5, 15\u003c/code\u003e\u003cbr\u003e\nFactors of \u003ccode\u003e20\u003c/code\u003e: \u003ccode\u003e1, 2, 4, 5, 10, 20\u003c/code\u003e\u003c/p\u003e\n\u003cp\u003eThe GCF of \u003ccode\u003e15\u003c/code\u003e and \u003ccode\u003e20\u003c/code\u003e is \u003ccode\u003e5\u003c/code\u003e, so Andi can make \u003ccode\u003e5\u003c/code\u003e identical bags.\u003c/p\u003e\n\u003ch3 id=\"3-patrol-schedule\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-patrol-schedule\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Patrol schedule\"\u003e\u003c/a\u003e3. Patrol schedule\u003c/h3\u003e\u003cp\u003eMr. Amin patrols every \u003ccode\u003e4\u003c/code\u003e days, Mr. Andi every \u003ccode\u003e6\u003c/code\u003e days, and Mr. Budi every \u003ccode\u003e8\u003c/code\u003e days. If they patrol together on \u003cstrong\u003eAugust 1, 2021\u003c/strong\u003e, when will they patrol together for the second time?\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n4 \u0026amp;= 2^2\\\\\n6 \u0026amp;= 2 \\times 3\\\\\n8 \u0026amp;= 2^3\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe LCM of \u003ccode\u003e4\u003c/code\u003e, \u003ccode\u003e6\u003c/code\u003e, and \u003ccode\u003e8\u003c/code\u003e is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2^3 \\times 3 = 24\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo they will meet again \u003ccode\u003e24\u003c/code\u003e days after \u003cstrong\u003eAugust 1, 2021\u003c/strong\u003e, which is \u003cstrong\u003eAugust 25, 2021\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eInteractive exercises for this section are available in the original source.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat completes this overview of how to find the LCM and GCF using lists of multiples, prime factorization, and repeated division. Choose the method that feels clearest, then reinforce it with practice.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eReference:\u003c/strong\u003e Mathematics textbook for Grade 7 junior secondary students, Ministry of Education and Culture of the Republic of Indonesia, 2016.\u003c/p\u003e\n", "language": "en" }, { "title" : "Education Calendar for Academic Year 2021-2022", "date_published" : "2021-07-27T06:59:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kalender-pendidikan-tahun-pelajaran/", "url" : "https://www.aantriono.com/en/kalender-pendidikan-tahun-pelajaran/", "summary": "SCurrently the school has entered a new academic year. Of course, there are many tasks carried out by teachers. As we know, the teacher\u0026rsquo;s job is to plan, implement, evaluate, assess and follow up on learning outcomes. Regarding planning learning, teachers need to make an effective week plan, syllabus, lesson plans, teaching materials, LKPD and so on. For timing, refer to the educational calendar made by the district or provincial education and culture office. The educational calendar is a time setting for students\u0026rsquo; learning activities during one academic year which includes the beginning of the academic year, effective learning weeks, effective learning time and holidays.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eSCurrently the school has entered a new academic year. Of course, there are many tasks carried out by teachers. As we know, the teacher\u0026rsquo;s job is to plan, implement, evaluate, assess and follow up on learning outcomes. Regarding planning learning, teachers need to make an effective week plan, syllabus, lesson plans, teaching materials, LKPD and so on. For timing, refer to the educational calendar made by the district or provincial education and culture office. The educational calendar is a time setting for students\u0026rsquo; learning activities during one academic year which includes the beginning of the academic year, effective learning weeks, effective learning time and holidays.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWUSG4B7OkACm4tRTi7fYlCpU5afQSpKnjhCqRUFv0GULuG9_sFkFFZYfErNFDC8oLiL9XQwnECZItz9zxiyEE-7eTvYjqGBUKF4BKCahB1moh_Kvu5ErHLfw3RpEuWdW7dHd15ftsgPaKKE8hfkTjHvL5UhvPsGehkBWc6UnIAVRZ03LP7aSMmx8_/s1280/kaldii.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"kalender pendidikan tahun pelajaran 2021/2022\" border=\"0\" data-original-height=\"400\" data-original-width=\"1280\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWUSG4B7OkACm4tRTi7fYlCpU5afQSpKnjhCqRUFv0GULuG9_sFkFFZYfErNFDC8oLiL9XQwnECZItz9zxiyEE-7eTvYjqGBUKF4BKCahB1moh_Kvu5ErHLfw3RpEuWdW7dHd15ftsgPaKKE8hfkTjHvL5UhvPsGehkBWc6UnIAVRZ03LP7aSMmx8_/w640-h200/kaldii.png\" title=\"kalender pendidikan\" width=\"640\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"lampung-province-education-calendar\" class=\"headerLink\"\u003e\n \u003ca href=\"#lampung-province-education-calendar\" class=\"header-mark\" aria-label=\"Direct link to section: Lampung Province Education Calendar\"\u003e\u003c/a\u003eLampung Province Education Calendar\u003c/h2\u003e\u003cp\u003ePreparation of Education Calendar and Number of Study Hours Effective in Schools is intended as a drafting guide School Education Calendar in implementing activities learning for schools for the 2021/2022 academic year. this guide determined based on the Decree of the Head of the Education Service and Culture of Lampung Province Number: 800/769/V.01/DP.1C/2021 March 19 2021 regarding Education Calendar and Number of Hours Effective Learning in Kindergarten/Outdoor Kindergarten Ordinary, Primary School/Special Primary School, Middle School First/Special Junior High School, Middle School Upper/Special High Schools and Middle Schools Vocational School Year 2021/2022 in Lampung Province.\u003c/p\u003e\n\u003cp\u003eBelow, I have embedded an educational calendar created by the Lampung Province Education and Culture Office.\n\u003ciframe allow=\"autoplay\" height=\"480\" src=\"https://drive.google.com/file/d/1QDHCTgFEipnpv2qBNw4IKY-mj7KLRwwn/preview\" width=\"660\"\u003e\u003c/iframe\u003e\n\u003c/p\u003e\n\u003ch2 id=\"educational-calendar-components\" class=\"headerLink\"\u003e\n \u003ca href=\"#educational-calendar-components\" class=\"header-mark\" aria-label=\"Direct link to section: Educational Calendar Components\"\u003e\u003c/a\u003eEducational Calendar Components\u003c/h2\u003e\u003cp\u003eThe start of the school year is the time when learning activities begin at the beginning of the school year in each educational unit. Effective learning weeks are the number of weeks of learning activities for each academic year in each educational unit. Effective learning time is the number of learning hours each week, including the number of learning hours for all subjects including local content, plus the number of hours for self-development activities. Holiday time is the time specified when scheduled learning activities are not held in the educational unit in question. Holiday time can take the form of mid-semester breaks, inter-semester breaks, end-of-school year holidays, religious holidays, public holidays including national holidays, and special holidays. After the school receives an educational calendar created by the district or provincial Education and Culture Office, the school then prepares an educational calendar at the educational unit level which will be used for one academic year. Of course, the educational calendar created is adjusted to the situation and conditions in the educational unit. Below is an example of an education unit level education calendar created by SMPN Purworejo Kec. Padangratu District. Central Lampung. Please download by clicking the following button.\u003c/p\u003e\n\u003cdiv class=\"downloadInfo\"\u003e\n \u003c!--[ Change data-text='...' atribute to add new file type ]--\u003e\n\u003cdiv class=\"fileName\"\u003e\n\u003c!--[ File name ]--\u003e\nfile_name.xlsx\n18kb\n \u003c!--[ Download link (change href='...' atribute to add link download) ]--\u003e\n \u003ca aria-label=\"Download\" class=\"button fileLink\" href=\"https://drive.google.com/file/d/1qlGTXD10TAIlGU-pjAbbL72PiOAI6_oh/view?usp=sharing\"\u003e\u003c/a\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThis is the article about the educational calendar for the 2021/2022 academic year. If there are errors, please correct them and if you have input, please write them here. I hope this is helpful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Fractions", "date_published" : "2021-07-17T13:20:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bilangan-pecahan/", "url" : "https://www.aantriono.com/en/bilangan-pecahan/", "summary": " IntroductionClass VII.A of a junior high school held an election for class president. The candidates were Danang, Fatimah, and Zalfa. After the votes were counted, Danang received 13\\dfrac{1}{3}31​ of Fatimah\u0026rsquo;s votes, while Fatimah received 34\\dfrac{3}{4}43​ of Zalfa\u0026rsquo;s votes. If there are 32 students in the class, how many votes did each candidate get?\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eClass VII.A of a junior high school held an election for class president. The candidates were Danang, Fatimah, and Zalfa. After the votes were counted, Danang received \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e of Fatimah\u0026rsquo;s votes, while Fatimah received \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e of Zalfa\u0026rsquo;s votes. If there are 32 students in the class, how many votes did each candidate get?\u003c/p\u003e\n\u003ch2 id=\"definition-of-fractions\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-of-fractions\" class=\"header-mark\" aria-label=\"Direct link to section: Definition of Fractions\"\u003e\u003c/a\u003eDefinition of Fractions\u003c/h2\u003e\u003cp\u003eQuoted from the \u003cem\u003eMathematics Encyclopedia\u003c/em\u003e by ST. Negoro and B. Harahap:\u003c/p\u003e\n\u003cp\u003eA fraction is a number that describes part of a whole, part of an area, part of an object, or part of a set.\u003c/p\u003e\n\u003cp\u003eLook at the following image.\u003c/p\u003e\n\u003cp\u003eIn a fraction written in the form \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{a}{b}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are integers and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb \\ne 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called the \u003cstrong\u003enumerator\u003c/strong\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called the \u003cstrong\u003edenominator\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eEquivalent fractions are fractions that have the same value. To obtain an equivalent fraction, the numerator and denominator can be multiplied or divided by the same nonzero number. Examples are \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{2}{2}, \\dfrac{2}{4}, \\dfrac{3}{6}, \\dfrac{4}{8}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{8}{16}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eTo compare two fractions with different denominators, first convert them into equivalent fractions with the same denominator.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eCompare the following pairs of fractions:\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{2}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{5}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ea. The denominators of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{2}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are 3 and 4. The LCM of 3 and 4 is 12.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{2}{3}=\\dfrac{8}{12} \\quad \\text{and} \\quad \\dfrac{3}{4}=\\dfrac{9}{12}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{8}{12}\u0026lt;\\dfrac{9}{12}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{2}{3}\u0026lt;\\dfrac{3}{4}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eb. The denominators of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{5}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are 5 and 7. The LCM of 5 and 7 is 35.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmn\u003e35\u003c/mn\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmn\u003e35\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{3}{5}=\\dfrac{21}{35} \\quad \\text{and} \\quad \\dfrac{5}{7}=\\dfrac{25}{35}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e35\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e35\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmn\u003e35\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmn\u003e35\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{21}{35}\u0026lt;\\dfrac{25}{35}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e35\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e35\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{3}{5}\u0026lt;\\dfrac{5}{7}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cstrong\u003eAlso read:\u003c/strong\u003e\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eIntegers\u003c/li\u003e\n\u003cli\u003eExponents\u003c/li\u003e\n\u003cli\u003eRatios\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eMrs. Intan has IDR 15,000.00. If she spends IDR 10,000.00, what fraction of her money has been spent?\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe ratio of the money spent to the total amount is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10,000\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e15,000\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n10{,}000 : 15{,}000 = 2 : 3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e000\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e000\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"decimal-fractions\" class=\"headerLink\"\u003e\n \u003ca href=\"#decimal-fractions\" class=\"header-mark\" aria-label=\"Direct link to section: Decimal Fractions\"\u003e\u003c/a\u003eDecimal Fractions\u003c/h2\u003e\u003cp\u003eDecimal fractions are fractions with denominators 10, 100, 1,000, and so on.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eConvert the following fractions to decimal form:\na. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{6}{25}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eConvert the following decimals to ordinary fractions:\na. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0.4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0.4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0.016\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0.016\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.016\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nc. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5.25\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5.25\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5.25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" 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style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.24\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{6}{25}=\\dfrac{6\\times 4}{25\\times 4}=\\dfrac{24}{100}=0.24\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan 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style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" 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style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e125\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nc. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5.25\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e525\u003c/mn\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e525\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5.25=\\dfrac{525}{100}=\\dfrac{525 : 25}{100 : 25}=\\dfrac{21}{4}=5\\dfrac{1}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5.25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e525\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e525\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"percent\" class=\"headerLink\"\u003e\n \u003ca href=\"#percent\" class=\"header-mark\" aria-label=\"Direct link to section: Percent\"\u003e\u003c/a\u003ePercent\u003c/h2\u003e\u003cp\u003ePercent is a fraction with a denominator of 100 and is denoted by \u003ccode\u003e%\u003c/code\u003e. To convert an ordinary fraction to percent form, you can:\u003c/p\u003e\n\u003cp\u003ea. change the original fraction into an equivalent fraction with denominator 100, or\u003cbr\u003e\nb. multiply the fraction by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e100\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eConvert the following fractions into percent form:\na. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{6}{25}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eConvert the following percent forms to ordinary fractions:\na. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e20\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e75\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e75\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e75%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nc. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e150\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e150\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e150%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{4}\\times 100\\%=25\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100%\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{6}{25}\\times 100\\%=24\\%\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100%\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24%\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e20\\%=\\dfrac{20}{100}=\\dfrac{1}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20%\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e75\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e75\u003c/mn\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e75\\%=\\dfrac{75}{100}=\\dfrac{3}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e75%\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e75\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nc. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e150\u003c/mn\u003e\u003cmi mathvariant=\"normal\"\u003e%\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e150\u003c/mn\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e150\\%=\\dfrac{150}{100}=\\dfrac{3}{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8056em;vertical-align:-0.0556em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e150%\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e150\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"permille\" class=\"headerLink\"\u003e\n \u003ca href=\"#permille\" class=\"header-mark\" aria-label=\"Direct link to section: Permille\"\u003e\u003c/a\u003ePermille\u003c/h2\u003e\u003cp\u003ePermille is a fraction with denominator 1,000. It is usually used to express salinity. For example, the salinity of the Red Sea reaches 40‰. That means every 1 kg of seawater contains 40 g of salt.\u003c/p\u003e\n\u003cp\u003eTo convert an ordinary fraction to permille form, you can:\u003c/p\u003e\n\u003cp\u003ea. change the fraction into an equivalent fraction with denominator 1,000, or\u003cbr\u003e\nb. multiply the fraction by 1,000‰.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eConvert the following fractions into permille form:\na. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{5}{8}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eConvert the following permille forms to ordinary fractions:\na. \n \n \\(25‰\\)\u003cbr\u003e\nb. \n \n \\(1{,}500‰\\)\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003ea. \n \n \\(\\dfrac{3}{5}=\\dfrac{3\\times 200}{5\\times 200}=\\dfrac{600}{1,000}=600‰\\)\u003cbr\u003e\nb. \n \n \\(\\dfrac{5}{8}\\times 1{,}000‰=625‰\\)\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003ea. \n \n \\(25‰=\\dfrac{25}{1,000}=\\dfrac{1}{40}\\)\u003cbr\u003e\nb. \n \n \\(1{,}500‰=\\dfrac{1,500}{1,000}=\\dfrac{3}{2}\\)\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"arithmetic-operations-with-fractions\" class=\"headerLink\"\u003e\n \u003ca href=\"#arithmetic-operations-with-fractions\" class=\"header-mark\" aria-label=\"Direct link to section: Arithmetic Operations with Fractions\"\u003e\u003c/a\u003eArithmetic Operations with Fractions\u003c/h2\u003e\u003ch3 id=\"addition\" class=\"headerLink\"\u003e\n \u003ca href=\"#addition\" class=\"header-mark\" aria-label=\"Direct link to section: Addition\"\u003e\u003c/a\u003eAddition\u003c/h3\u003e\u003cp\u003eFractions with the same denominator can be added by adding only the numerators, while the denominator stays the same. In general:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{a}{c}+\\dfrac{b}{c}=\\dfrac{a+b}{c}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf the fractions have different denominators, first make the denominators equal by finding the LCM.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eCalculate the sum of the following fractions:\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{7}+\\dfrac{1}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{4}+\\dfrac{2}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{7}+\\dfrac{1}{7}=\\dfrac{3+1}{7}=\\dfrac{4}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eb. The LCM of 4 and 3 is 12, so:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{1}{4}+\\dfrac{2}{3}=\\dfrac{3}{12}+\\dfrac{8}{12}=\\dfrac{11}{12}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"subtraction\" class=\"headerLink\"\u003e\n \u003ca href=\"#subtraction\" class=\"header-mark\" aria-label=\"Direct link to section: Subtraction\"\u003e\u003c/a\u003eSubtraction\u003c/h3\u003e\u003cp\u003eFractions with the same denominator can be subtracted by subtracting the numerators, while the denominator stays the same. In general:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{a}{c}-\\dfrac{b}{c}=\\dfrac{a-b}{c}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf the denominators are different, first make them equal.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDetermine the results of the following subtractions:\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{5}{7}-\\dfrac{2}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{4}-\\dfrac{1}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nc. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\\dfrac{2}{3}-2\\dfrac{1}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{5}{7}-\\dfrac{2}{7}=\\dfrac{5-2}{7}=\\dfrac{3}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eb. The LCM of 4 and 5 is 20, so:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{3}{4}-\\dfrac{1}{5}=\\dfrac{15}{20}-\\dfrac{4}{20}=\\dfrac{11}{20}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ec. The LCM of 3 and 5 is 15, so:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n4\\dfrac{2}{3}-2\\dfrac{1}{5}\n\u0026amp;=(4-2)+\\left(\\dfrac{2}{3}-\\dfrac{1}{5}\\right)\\\\\n\u0026amp;=2+\\left(\\dfrac{10}{15}-\\dfrac{3}{15}\\right)\\\\\n\u0026amp;=2+\\dfrac{7}{15}\\\\\n\u0026amp;=2\\dfrac{7}{15}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:10.0149em;vertical-align:-4.7575em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.2575em;\"\u003e\u003cspan style=\"top:-7.2575em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.986em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.3215em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7575em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.2575em;\"\u003e\u003cspan style=\"top:-7.2575em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.986em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.3215em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7575em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"multiplication\" class=\"headerLink\"\u003e\n \u003ca href=\"#multiplication\" class=\"header-mark\" aria-label=\"Direct link to section: Multiplication\"\u003e\u003c/a\u003eMultiplication\u003c/h3\u003e\u003cp\u003eMultiplying fractions means multiplying a number by a fraction. It may involve a whole number and a fraction, or two fractions. In general:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{a}{b}\\times \\dfrac{c}{d}=\\dfrac{a\\times c}{b\\times d}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0297em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDetermine the products of the following fractions:\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{4}\\times \\dfrac{7}{10}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{5}{6}\\times \\dfrac{6}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" 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displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmn\u003e42\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e42\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\dfrac{5}{6}\\times \\dfrac{6}{7}\n\u0026amp;=\\dfrac{5\\times 6}{6\\times 7}\\\\\n\u0026amp;=\\dfrac{30}{42}\\\\\n\u0026amp;=\\dfrac{30 : 6}{42 : 6}\\\\\n\u0026amp;=\\dfrac{5}{7}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:9.3131em;vertical-align:-4.4065em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan 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class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5158em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.2083em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.0991em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4065em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.9065em;\"\u003e\u003cspan style=\"top:-6.9065em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5158em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e42\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.2083em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e42\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.0991em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4065em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe numerator and denominator are divided by the GCF of 30 and 42, namely 6.\u003c/p\u003e\n\u003ch3 id=\"division\" class=\"headerLink\"\u003e\n \u003ca href=\"#division\" class=\"header-mark\" aria-label=\"Direct link to section: Division\"\u003e\u003c/a\u003eDivision\u003c/h3\u003e\u003cp\u003eDividing by a fraction is the same as multiplying by the reciprocal of the divisor. In general:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e÷\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\dfrac{a}{b}\\div \\dfrac{c}{d}=\\dfrac{a}{b}\\times \\dfrac{d}{c}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e÷\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDetermine the results of the following divisions:\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e÷\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{2}\\div \\dfrac{3}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e÷\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nb. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e÷\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2 \\div \\dfrac{1}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e÷\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nc. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e÷\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{4}{5}\\div 4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e÷\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ea. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e÷\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{2}\\div \\dfrac{3}{4}=\\dfrac{1}{2}\\times \\dfrac{4}{3}=\\dfrac{4}{6}=\\dfrac{2}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003ec. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e÷\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{4}{5}\\div 4=\\dfrac{4}{5}\\times \\dfrac{1}{4}=\\dfrac{4\\times 1}{5\\times 4}=\\dfrac{1}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e÷\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003ch2 id=\"sequence-of-calculation-operations\" class=\"headerLink\"\u003e\n \u003ca href=\"#sequence-of-calculation-operations\" class=\"header-mark\" aria-label=\"Direct link to section: Sequence of Calculation Operations\"\u003e\u003c/a\u003eSequence of Calculation Operations\u003c/h2\u003e\u003cp\u003eIf you find a mathematical sentence that mixes add, subtract, multiply, divide and exponent operations, then we use the following rules for the order of arithmetic operations.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eComplete the power operation.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eComplete the multiply and divide operations.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eComplete the add and subtract operations. Operations in parentheses are performed first.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eTo evaluate your understanding of fractions, try solving the examples in each section again without looking at the solutions first. Then compare your work with the discussion in the article.\u003c/p\u003e\n\u003cp\u003ePay special attention to these skills:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ecomparing fractions,\u003c/li\u003e\n\u003cli\u003econverting fractions to decimals, percentages, and permille,\u003c/li\u003e\n\u003cli\u003eperforming arithmetic operations with fractions,\u003c/li\u003e\n\u003cli\u003eapplying the correct order of operations.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat concludes this discussion of fractions. If you find any part that needs correction, please leave a note in the comments.\u003c/p\u003e\n", "language": "en" }, { "title" : "Radical Forms and Fractional Exponents", "date_published" : "2021-07-13T13:58:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/07/bentuk-akar-dan-pangkat-pecahan/", "url" : "https://www.aantriono.com/2021/07/bentuk-akar-dan-pangkat-pecahan/", "summary": " IntroductionIrrational numbers are closely related to radical forms. Numbers such as 2\\sqrt{2}2​, 3\\sqrt{3}3​, and 5\\sqrt{5}5​ are examples of irrational numbers written in radical form.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIrrational numbers are closely related to radical forms. Numbers such as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are examples of irrational numbers written in radical form.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eRational numbers are numbers that can be expressed as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{a}{b}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are integers and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb \\ne 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eIrrational numbers are numbers that cannot be expressed as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{a}{b}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are integers and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb \\ne 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eExamples of rational numbers: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-\\dfrac{2}{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3}{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eExamples of irrational numbers: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eWhen evaluated with a calculator, irrational numbers appear as non-terminating and non-repeating decimals. For example:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1.414213562\u003c/mn\u003e\u003cmo\u003e…\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\sqrt{2}=1.414213562\\ldots\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0839em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1.414213562\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe union of rational and irrational numbers is called the set of \u003cstrong\u003ereal numbers\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"understanding-radical-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#understanding-radical-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Understanding Radical Forms\"\u003e\u003c/a\u003eUnderstanding Radical Forms\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtigZx1MIhWOhnBE8Mtzk6nL2zgh7lipGjdRUt_m0Xz4bjzbipjuIVesv7MKxftaMPu_O_ea6ZO-D-fD8zqe33m175tusrnHnKJE2dg9hQv7HFehBIWkkhMM9_5F9PpDPB4IyhXiq_sew/w640-h297/Screenshot+2021-07-13+21.01.59.png' alt=\"Radical forms and fractional exponents\" \u003e\u003c/p\u003e\n\u003cp\u003eConsider the following examples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{4}=\\sqrt{2^2}=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0849em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9551em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9151em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0849em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{9}=\\sqrt{3^2}=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0849em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9551em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9151em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0849em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmsup\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{16}=\\sqrt{4^2}=4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0849em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9551em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9151em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0849em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThese examples follow the definition:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\sqrt{a^2}=a\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0361em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0039em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9639em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0361em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003efor positive real \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eHowever, for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e there is no rational number whose square is 2. Numbers of this kind are called \u003cstrong\u003eradical forms\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\sqrt{2},\\ \\sqrt{3},\\ \\sqrt{5}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1505em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"algebraic-operations-on-radical-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#algebraic-operations-on-radical-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Algebraic Operations on Radical Forms\"\u003e\u003c/a\u003eAlgebraic Operations on Radical Forms\u003c/h2\u003e\u003ch3 id=\"1-addition-and-subtraction\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-addition-and-subtraction\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Addition and Subtraction\"\u003e\u003c/a\u003e1. Addition and Subtraction\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsqrt\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmsqrt\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsqrt\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na\\sqrt{c}+b\\sqrt{c}=(a+b)\\sqrt{c}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1908em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8492em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8092em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1908em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1908em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8492em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8092em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1908em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0992em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8492em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8092em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1908em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsqrt\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmsqrt\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsqrt\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na\\sqrt{c}-b\\sqrt{c}=(a-b)\\sqrt{c}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1908em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8492em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8092em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1908em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1908em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8492em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8092em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1908em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0992em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8492em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8092em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1908em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi mathvariant=\"double-struck\"\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea,b,c \\in \\mathbb{R}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6889em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathbb\"\u003eR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e≥\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec \\ge 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7719em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch4 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eSimplify:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\\sqrt{3}-2\\sqrt{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" 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vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath 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xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e15\\sqrt{7}-25\\sqrt{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" 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id=\"answer\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo 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href=\"#2-multiplication\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Multiplication\"\u003e\u003c/a\u003e2. 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style=\"height:0.8304em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo\u003e≥\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed \\ge 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8304em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eFind:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\sqrt{3}\\times 5\\sqrt{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{8}\\times \\sqrt{12}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2\\sqrt{5})(2\\sqrt{2})-3\\sqrt{2}(3\\sqrt{5}-2\\sqrt{3})\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1572em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan 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xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-1\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" 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displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n2\\sqrt{3}\\times 5\\sqrt{2}\n\u0026amp;=(2\\times 5)\\sqrt{3\\times 2}\\\\\n\u0026amp;=10\\sqrt{6}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.2322em;vertical-align:-1.3661em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.8661em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath 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style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3661em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-division\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-division\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Division\"\u003e\u003c/a\u003e3. Division\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmsqrt\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmsqrt\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mfrac\u003e\u003cmsqrt\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation 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class=\"vlist\" style=\"height:0.9322em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8922em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8003em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.7603em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.5047em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.4em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:2.48em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"2.48em\" viewBox=\"0 0 400000 2592\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8953em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea,b,p,q\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are real numbers and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e≥\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\ge 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7719em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e≥\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb \\ge 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8304em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eFind:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmsqrt\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msqrt\u003e\u003cmsqrt\u003e\u003cmn\u003e28\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{\\sqrt{7}}{\\sqrt{28}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.5142em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5842em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e28\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{10\\sqrt{8}}{5\\sqrt{2}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.5142em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5842em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-2\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmsqrt\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msqrt\u003e\u003cmsqrt\u003e\u003cmn\u003e28\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e28\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{\\sqrt{7}}{\\sqrt{28}}\n=\\sqrt{\\frac{7}{28}}\n=\\sqrt{\\frac{1}{4}}\n=\\frac{1}{2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.5142em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5842em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e28\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8656em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8256em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.5142em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5842em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"rationalizing-the-denominator\" class=\"headerLink\"\u003e\n \u003ca href=\"#rationalizing-the-denominator\" class=\"header-mark\" aria-label=\"Direct link to section: Rationalizing the Denominator\"\u003e\u003c/a\u003eRationalizing the Denominator\u003c/h2\u003e\u003cp\u003eRationalizing the denominator means changing a fraction whose denominator is irrational into one whose denominator is rational.\u003c/p\u003e\n\u003ch3 id=\"1-rationalizing\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-rationalizing\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Rationalizing ab\\dfrac{a}{\\sqrt{b}}b​a​\"\u003e\u003c/a\u003e1. 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style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.1778em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9322em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8922em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1078em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan 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class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9322em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8922em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1078em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eRationalize:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{4}{\\sqrt{5}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2514em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cmsqrt\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{-6}{\\sqrt{7}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2514em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmsqrt\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{\\sqrt{3}}{\\sqrt{6}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.5142em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5842em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-3\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" 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class=\"mord\"\u003e18\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-rationalizing\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-rationalizing\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Rationalizing ap±q\\dfrac{a}{p\\pm\\sqrt{q}}p±q​a​\"\u003e\u003c/a\u003e2. Rationalizing \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{a}{p\\pm\\sqrt{q}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1305em;vertical-align:-1.0229em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7031em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6631em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3369em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0229em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h3\u003e\u003cp\u003eFractions with denominators of the form \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep+\\sqrt{q}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.3369em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7031em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6631em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3369em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep-\\sqrt{q}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.3369em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7031em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6631em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3369em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are rationalized by multiplying by the conjugate of the denominator.\u003c/p\u003e\n\u003ch4 id=\"example-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-4\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eRationalize:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{8}{3+\\sqrt{5}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2514em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{3\\sqrt{3}}{3-\\sqrt{6}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.5142em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5842em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-4\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\frac{8}{3+\\sqrt{5}}\n\u0026amp;=\\frac{8}{3+\\sqrt{5}}\\times \\frac{3-\\sqrt{5}}{3-\\sqrt{5}}\\\\\n\u0026amp;=\\frac{8(3-\\sqrt{5})}{9-5}\\\\\n\u0026amp;=\\frac{8(3-\\sqrt{5})}{4}\\\\\n\u0026amp;=2(3-\\sqrt{5})\\\\\n\u0026amp;=6-2\\sqrt{5}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:11.2702em;vertical-align:-5.3851em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.8851em;\"\u003e\u003cspan style=\"top:-7.8851em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.0709em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.4173em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.4752em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.1409em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan 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class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.3328em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.9489em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.1931em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-rationalizing\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-rationalizing\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Rationalizing ca±b\\dfrac{c}{\\sqrt{a}\\pm\\sqrt{b}}a​±b​c​\"\u003e\u003c/a\u003e3. Rationalizing \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/msqrt\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{c}{\\sqrt{a}\\pm\\sqrt{b}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1695em;vertical-align:-1.0619em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.1778em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8003em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.7603em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2397em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9322em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8922em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1078em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0619em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h3\u003e\u003cp\u003eThis is done by multiplying by the conjugate of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/msqrt\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{a}\\pm\\sqrt{b}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.2397em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8003em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.7603em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2397em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1078em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9322em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8922em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1078em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch4 id=\"example-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-5\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eRationalize:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{8}{\\sqrt{5}+\\sqrt{2}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2514em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan 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scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{5}{3\\sqrt{2}-\\sqrt{3}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2514em;vertical-align:-0.93em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.2028em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.93em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-5\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\frac{8}{\\sqrt{5}+\\sqrt{2}}\n\u0026amp;=\\frac{8}{\\sqrt{5}+\\sqrt{2}}\\times \\frac{\\sqrt{5}-\\sqrt{2}}{\\sqrt{5}-\\sqrt{2}}\\\\\n\u0026amp;=\\frac{8(\\sqrt{5}-\\sqrt{2})}{5-2}\\\\\n\u0026amp;=\\frac{8}{3}(\\sqrt{5}-\\sqrt{2})\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" 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class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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style=\"top:-3.3234em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5842em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5842em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e18\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\frac{5}{3\\sqrt{2}-\\sqrt{3}}\n\u0026amp;=\\frac{5}{3\\sqrt{2}-\\sqrt{3}}\\times \\frac{3\\sqrt{2}+\\sqrt{3}}{3\\sqrt{2}+\\sqrt{3}}\\\\\n\u0026amp;=\\frac{5(3\\sqrt{2}+\\sqrt{3})}{18-3}\\\\\n\u0026amp;=\\frac{5(3\\sqrt{2}+\\sqrt{3})}{15}\\\\\n\u0026amp;=\\frac{1}{3}(3\\sqrt{2}+\\sqrt{3})\\\\\n\u0026amp;=\\sqrt{2}+\\frac{1}{3}\\sqrt{3}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" 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xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 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style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{10}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e can be written using fractional exponents:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsqrt\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsqrt\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\sqrt{2}=2^{\\frac{1}{2}},\\quad\n\\sqrt{3}=3^{\\frac{1}{2}},\\quad\n\\sqrt{5}=5^{\\frac{1}{2}},\\quad\n\\sqrt{10}=10^{\\frac{1}{2}}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0839em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1985em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.004em;\"\u003e\u003cspan style=\"top:-3.413em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1985em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.004em;\"\u003e\u003cspan style=\"top:-3.413em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1985em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.004em;\"\u003e\u003cspan style=\"top:-3.413em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.004em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.004em;\"\u003e\u003cspan style=\"top:-3.413em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe general relationship is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmfrac\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmroot\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msup\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mroot\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^{\\frac{m}{n}}=\\sqrt[n]{a^m}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8971em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8971em;\"\u003e\u003cspan style=\"top:-3.413em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6915em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1109em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7062em;\"\u003e\u003cspan style=\"top:-2.9909em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9291em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5904em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8891em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1109em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e≥\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\ge 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7719em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em,n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e positive integers.\u003c/p\u003e\n\u003cp\u003eFor a review, see the material on \u003ca href=\"https://www.aantriono.com/2020/07/bilangan-berpangkat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eexponents\u003c/a\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-6\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-6\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eExpress the following radical forms as exponents:\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmroot\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mroot\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt[3]{4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7869em;\"\u003e\u003cspan style=\"top:-2.9647em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmroot\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mroot\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt[5]{16}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7869em;\"\u003e\u003cspan style=\"top:-2.9647em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003eExpress the following fractional exponents as radicals:\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^{\\frac{2}{3}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.954em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.954em;\"\u003e\u003cspan style=\"top:-3.363em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2^{\\frac{3}{5}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.954em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.954em;\"\u003e\u003cspan style=\"top:-3.363em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-6\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-6\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eAs exponents:\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmroot\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mroot\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmroot\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mroot\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt[3]{4}=\\sqrt[3]{2^2}=2^{\\frac{2}{3}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7869em;\"\u003e\u003cspan style=\"top:-2.9647em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0849em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-3.0221em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9551em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9151em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.954em;\"\u003e\u003cspan style=\"top:-3.363em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mopen nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-2.656em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.2255em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.384em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.344em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter sizing reset-size3 size6\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmroot\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mroot\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmroot\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mroot\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt[5]{16}=\\sqrt[5]{2^4}=2^{\\frac{4}{5}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7869em;\"\u003e\u003cspan style=\"top:-2.9647em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0849em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8443em;\"\u003e\u003cspan style=\"top:-3.0221em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9551em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9151em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg 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style=\"height:0.0849em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.954em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.954em;\"\u003e\u003cspan style=\"top:-3.363em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan 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xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmroot\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mroot\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmroot\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mroot\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^{\\frac{2}{3}}=\\sqrt[3]{3^2}=\\sqrt[3]{9}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.954em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan 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mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9151em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7869em;\"\u003e\u003cspan style=\"top:-2.9647em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mfrac\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmroot\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mroot\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmroot\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mroot\u003e\u003c/mrow\u003e\u003cannotation 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d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0849em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"root\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7869em;\"\u003e\u003cspan style=\"top:-2.9647em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eAfter studying radical forms and fractional exponents, do the evaluation by:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eclicking the correct answer,\u003c/li\u003e\n\u003cli\u003eclicking \u003ccode\u003eNext\u003c/code\u003e to move to the next question,\u003c/li\u003e\n\u003cli\u003eclicking \u003ccode\u003eSubmit\u003c/code\u003e after all questions are done,\u003c/li\u003e\n\u003cli\u003echecking the score at the bottom,\u003c/li\u003e\n\u003cli\u003ecalculating the final mark as score divided by maximum score times 100.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat concludes this material on radical forms and fractional exponents.\u003c/p\u003e\n", "language": "en" }, { "title" : "Grade 7 Mathematics Materials", "date_published" : "2021-07-11T15:04:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2020/07/materi-matematika-kelas-vii/", "url" : "https://www.aantriono.com/2020/07/materi-matematika-kelas-vii/", "summary": " IntroductionCongratulations to those of you who have graduated from elementary school and entered junior high school. The material you studied before will be very useful at this level, including mathematics. This article summarizes Grade 7 mathematics materials based on the 2013 curriculum revision.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eCongratulations to those of you who have graduated from elementary school and entered junior high school. The material you studied before will be very useful at this level, including mathematics. This article summarizes Grade 7 mathematics materials based on the 2013 curriculum revision.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPMkX0FuiaWd37dAIqZ3-VlRZSJ8HpByFkBa3fzcYemv9V5-YVQ2GXJw0H2MoAEc1Qx4XVGWKJA4i1_9fbp_oyWmBEU3cNjIZrBGfiQg3mSfqBpX-eRFTmOVpN5g-zZEOd4gCRGPewvNv9bHAHhOwZDosX98dMFOQrgsBSO-P7-Zy58TPW-SnHWDkO/w640-h200/mtk.png' alt=\"Grade 7 mathematics materials\" \u003e\u003c/p\u003e\n\u003cp\u003eThe material includes:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eNumbers\u003c/li\u003e\n\u003cli\u003eSets\u003c/li\u003e\n\u003cli\u003eAlgebraic forms\u003c/li\u003e\n\u003cli\u003eLinear equations and inequalities in one variable\u003c/li\u003e\n\u003cli\u003eComparison\u003c/li\u003e\n\u003cli\u003eSocial arithmetic\u003c/li\u003e\n\u003cli\u003eLines and angles\u003c/li\u003e\n\u003cli\u003eQuadrilaterals and triangles\u003c/li\u003e\n\u003cli\u003eData presentation\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"semester-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#semester-1\" class=\"header-mark\" aria-label=\"Direct link to section: Semester 1\"\u003e\u003c/a\u003eSemester 1\u003c/h3\u003e\u003ch4 id=\"chapter-1-numbers\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-1-numbers\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 1 Numbers\"\u003e\u003c/a\u003eChapter 1 Numbers\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eCompare whole numbers and fractions\u003c/li\u003e\n\u003cli\u003eOrder whole numbers and fractions\u003c/li\u003e\n\u003cli\u003eOperations on whole numbers and fractions\u003c/li\u003e\n\u003cli\u003eChange the form of fractions\u003c/li\u003e\n\u003cli\u003eExpress numbers as positive whole powers\u003c/li\u003e\n\u003cli\u003eLeast common multiple and greatest common factor\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"chapter-2-sets\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-2-sets\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 2 Sets\"\u003e\u003c/a\u003eChapter 2 Sets\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eState a set\u003c/li\u003e\n\u003cli\u003eSubsets, empty set, and universal set\u003c/li\u003e\n\u003cli\u003eComplement of a set\u003c/li\u003e\n\u003cli\u003eRelationships between sets\u003c/li\u003e\n\u003cli\u003eSet operations\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"chapter-3-algebraic-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-3-algebraic-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 3 Algebraic Forms\"\u003e\u003c/a\u003eChapter 3 Algebraic Forms\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eCoefficients, variables, constants, and terms\u003c/li\u003e\n\u003cli\u003eOperations on algebraic forms\u003c/li\u003e\n\u003cli\u003eSimplifying algebraic forms\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"chapter-4-linear-equations-and-inequalities-in-one-variable\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-4-linear-equations-and-inequalities-in-one-variable\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 4 Linear Equations and Inequalities in One Variable\"\u003e\u003c/a\u003eChapter 4 Linear Equations and Inequalities in One Variable\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eStatements\u003c/li\u003e\n\u003cli\u003eOpen sentences\u003c/li\u003e\n\u003cli\u003eSolving linear equations in one variable\u003c/li\u003e\n\u003cli\u003eSolving linear inequalities in one variable\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"semester-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#semester-2\" class=\"header-mark\" aria-label=\"Direct link to section: Semester 2\"\u003e\u003c/a\u003eSemester 2\u003c/h3\u003e\u003ch4 id=\"chapter-5-comparison\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-5-comparison\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 5 Comparison\"\u003e\u003c/a\u003eChapter 5 Comparison\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eComparing two quantities\u003c/li\u003e\n\u003cli\u003eDirect proportion\u003c/li\u003e\n\u003cli\u003eInverse proportion\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"chapter-6-social-arithmetic\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-6-social-arithmetic\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 6 Social Arithmetic\"\u003e\u003c/a\u003eChapter 6 Social Arithmetic\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eSelling price and buying price\u003c/li\u003e\n\u003cli\u003eProfit, loss, and break-even\u003c/li\u003e\n\u003cli\u003eProfit and loss percentages\u003c/li\u003e\n\u003cli\u003eDiscount\u003c/li\u003e\n\u003cli\u003eTax\u003c/li\u003e\n\u003cli\u003eGross, tare, and net\u003c/li\u003e\n\u003cli\u003eSimple interest\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"chapter-7-lines-and-angles\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-7-lines-and-angles\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 7 Lines and Angles\"\u003e\u003c/a\u003eChapter 7 Lines and Angles\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eLines\u003c/li\u003e\n\u003cli\u003ePosition of lines\u003c/li\u003e\n\u003cli\u003eDividing a line\u003c/li\u003e\n\u003cli\u003eComparing line segments\u003c/li\u003e\n\u003cli\u003eDefinition of angles\u003c/li\u003e\n\u003cli\u003eTypes of angles\u003c/li\u003e\n\u003cli\u003eRelationships between angles\u003c/li\u003e\n\u003cli\u003eConstructing angles\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"chapter-8-quadrilaterals-and-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-8-quadrilaterals-and-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 8 Quadrilaterals and Triangles\"\u003e\u003c/a\u003eChapter 8 Quadrilaterals and Triangles\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eDefinition of quadrilaterals and triangles\u003c/li\u003e\n\u003cli\u003eTypes and properties of plane figures\u003c/li\u003e\n\u003cli\u003ePerimeter and area of quadrilaterals and triangles\u003c/li\u003e\n\u003cli\u003eEstimating the area of irregular plane figures\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"chapter-9-data-presentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-9-data-presentation\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 9 Data Presentation\"\u003e\u003c/a\u003eChapter 9 Data Presentation\u003c/h4\u003e\u003cul\u003e\n\u003cli\u003eTypes of data\u003c/li\u003e\n\u003cli\u003eTables\u003c/li\u003e\n\u003cli\u003eLine charts\u003c/li\u003e\n\u003cli\u003eBar charts\u003c/li\u003e\n\u003cli\u003ePie charts\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eYou can study this material at school, from textbooks, or online. On this blog, the material is presented as a summary, supported by images, tables, examples, GeoGebra simulations, and evaluations.\u003c/p\u003e\n\u003cp\u003eI hope this summary helps you understand Grade 7 mathematics more easily.\u003c/p\u003e\n", "language": "en" }, { "title" : "Grade 9 Mathematics Materials", "date_published" : "2021-07-11T14:01:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2020/07/materi-matematika-kelas-9-smpmts/", "url" : "https://www.aantriono.com/2020/07/materi-matematika-kelas-9-smpmts/", "summary": " Introduction\nThis page summarizes Grade 9 mathematics materials for junior secondary school based on the revised 2013 curriculum. In general, the material is divided into 6 main chapters.\nMain Chapters Exponents and Radical Forms Quadratic Equations Quadratic Functions Geometric Transformations Similarity and Congruence Curved Surface Solids Detailed Topics Chapter 1. Exponents and Radical Forms Integers raised to integer powers Fractions raised to integer powers Radical forms and fractional exponents Algebraic operations on radical forms Rationalizing denominators Chapter 2. Quadratic Equations Definition of quadratic equations Roots of quadratic equations Constructing quadratic equations Chapter 3. Quadratic Functions Quadratic functions Axis of symmetry and discriminant Graphing quadratic functions Applications of quadratic functions Chapter 4. Geometric Transformations Translation Reflection Rotation Dilation Chapter 5. Similarity and Congruence Similarity of plane figures Similar triangles Congruent triangles Finding side lengths and angle measures in congruent triangles Chapter 6. Curved Surface Solids Surface area of cylinders, cones, and spheres Volume of cylinders, cones, and spheres Changes in the volume of cylinders, cones, and spheres NoteThe materials on this blog are presented as summaries, illustrations, example problems, GeoGebra simulations, and interactive evaluations. They are intended to support understanding, not replace the main textbook.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcR2-tw6MXN6HHXEfKRxxVkIDdjXnnpyf2paQU63CwSRQ0an-yqbDoqL0bR7AqCsmmH4qT-2UphUfcr55hYlQD747oBC3aQBqwGzvfwwHXbw_juw6Y-iRKLoCJjvOUgFpH5ARrVwjAaoPs-hddbd-uQidJxAYkBtobCHSBroOpmKh3sCxEf9BfvLa9/w640-h200/9.png' alt=\"Grade 9 mathematics materials\" \u003e\u003c/p\u003e\n\u003cp\u003eThis page summarizes Grade 9 mathematics materials for junior secondary school based on the revised 2013 curriculum. In general, the material is divided into 6 main chapters.\u003c/p\u003e\n\u003ch2 id=\"main-chapters\" class=\"headerLink\"\u003e\n \u003ca href=\"#main-chapters\" class=\"header-mark\" aria-label=\"Direct link to section: Main Chapters\"\u003e\u003c/a\u003eMain Chapters\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eExponents and Radical Forms\u003c/li\u003e\n\u003cli\u003eQuadratic Equations\u003c/li\u003e\n\u003cli\u003eQuadratic Functions\u003c/li\u003e\n\u003cli\u003eGeometric Transformations\u003c/li\u003e\n\u003cli\u003eSimilarity and Congruence\u003c/li\u003e\n\u003cli\u003eCurved Surface Solids\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"detailed-topics\" class=\"headerLink\"\u003e\n \u003ca href=\"#detailed-topics\" class=\"header-mark\" aria-label=\"Direct link to section: Detailed Topics\"\u003e\u003c/a\u003eDetailed Topics\u003c/h2\u003e\u003ch3 id=\"chapter-1-exponents-and-radical-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-1-exponents-and-radical-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 1. Exponents and Radical Forms\"\u003e\u003c/a\u003eChapter 1. Exponents and Radical Forms\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eIntegers raised to integer powers\u003c/li\u003e\n\u003cli\u003eFractions raised to integer powers\u003c/li\u003e\n\u003cli\u003eRadical forms and fractional exponents\u003c/li\u003e\n\u003cli\u003eAlgebraic operations on radical forms\u003c/li\u003e\n\u003cli\u003eRationalizing denominators\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"chapter-2-quadratic-equations\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-2-quadratic-equations\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 2. Quadratic Equations\"\u003e\u003c/a\u003eChapter 2. Quadratic Equations\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eDefinition of quadratic equations\u003c/li\u003e\n\u003cli\u003eRoots of quadratic equations\u003c/li\u003e\n\u003cli\u003eConstructing quadratic equations\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"chapter-3-quadratic-functions\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-3-quadratic-functions\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 3. Quadratic Functions\"\u003e\u003c/a\u003eChapter 3. Quadratic Functions\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eQuadratic functions\u003c/li\u003e\n\u003cli\u003eAxis of symmetry and discriminant\u003c/li\u003e\n\u003cli\u003eGraphing quadratic functions\u003c/li\u003e\n\u003cli\u003eApplications of quadratic functions\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"chapter-4-geometric-transformations\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-4-geometric-transformations\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 4. Geometric Transformations\"\u003e\u003c/a\u003eChapter 4. Geometric Transformations\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eTranslation\u003c/li\u003e\n\u003cli\u003eReflection\u003c/li\u003e\n\u003cli\u003eRotation\u003c/li\u003e\n\u003cli\u003eDilation\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"chapter-5-similarity-and-congruence\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-5-similarity-and-congruence\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 5. Similarity and Congruence\"\u003e\u003c/a\u003eChapter 5. Similarity and Congruence\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eSimilarity of plane figures\u003c/li\u003e\n\u003cli\u003eSimilar triangles\u003c/li\u003e\n\u003cli\u003eCongruent triangles\u003c/li\u003e\n\u003cli\u003eFinding side lengths and angle measures in congruent triangles\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"chapter-6-curved-surface-solids\" class=\"headerLink\"\u003e\n \u003ca href=\"#chapter-6-curved-surface-solids\" class=\"header-mark\" aria-label=\"Direct link to section: Chapter 6. Curved Surface Solids\"\u003e\u003c/a\u003eChapter 6. Curved Surface Solids\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eSurface area of cylinders, cones, and spheres\u003c/li\u003e\n\u003cli\u003eVolume of cylinders, cones, and spheres\u003c/li\u003e\n\u003cli\u003eChanges in the volume of cylinders, cones, and spheres\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"note\" class=\"headerLink\"\u003e\n \u003ca href=\"#note\" class=\"header-mark\" aria-label=\"Direct link to section: Note\"\u003e\u003c/a\u003eNote\u003c/h2\u003e\u003cp\u003eThe materials on this blog are presented as summaries, illustrations, example problems, GeoGebra simulations, and interactive evaluations. They are intended to support understanding, not replace the main textbook.\u003c/p\u003e\n\u003cp\u003eTo begin with Chapter 1, please open:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ca href=\"https://www.aantriono.com/2020/07/bilangan-berpangkat/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eExponents\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eI hope this summary helps students study mathematics more easily.\u003c/p\u003e\n", "language": "en" }, { "title" : "Integers", "date_published" : "2021-07-01T07:17:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bilangan-bulat/", "url" : "https://www.aantriono.com/en/bilangan-bulat/", "summary": " A. IntroductionIntegers are used to describe many everyday situations, such as temperature, elevation, profit and loss, or positions on a number line. On a thermometer, temperatures below zero are written as negative numbers, while temperatures above zero are written as positive numbers.\nNegative integers are to the left of zero. Zero is in the middle. Positive integers are to the right of zero. Examples of integers: ..., -3, -2, -1, 0, 1, 2, 3, ...\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch4 id=\"a-introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-introduction\" class=\"header-mark\" aria-label=\"Direct link to section: A. Introduction\"\u003e\u003c/a\u003eA. Introduction\u003c/h4\u003e\u003cp\u003eIntegers are used to describe many everyday situations, such as temperature, elevation, profit and loss, or positions on a number line. On a thermometer, temperatures below zero are written as negative numbers, while temperatures above zero are written as positive numbers.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eNegative integers are to the left of zero.\u003c/li\u003e\n\u003cli\u003eZero is in the middle.\u003c/li\u003e\n\u003cli\u003ePositive integers are to the right of zero.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExamples of integers: \u003ccode\u003e..., -3, -2, -1, 0, 1, 2, 3, ...\u003c/code\u003e\u003c/p\u003e\n\u003ch4 id=\"b-definition\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-definition\" class=\"header-mark\" aria-label=\"Direct link to section: B. Definition\"\u003e\u003c/a\u003eB. Definition\u003c/h4\u003e\u003cp\u003eIntegers are the set of negative integers, zero, and positive integers. The standard notation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nZ = \\{..., -3, -2, -1, 0, 1, 2, 3, ...\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eZ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e...\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e...\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eOn a number line, numbers become larger as you move to the right. Because of that, the symbols \u003ccode\u003e\u0026lt;\u003c/code\u003e and \u003ccode\u003e\u0026gt;\u003c/code\u003e are used to compare integers.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n3 \u0026amp;\u0026gt; -1\\\\\n-4 \u0026amp;\u0026lt; 2\\\\\n-3 \u0026amp;\u0026lt; -1\\\\\n7 \u0026amp;\u0026gt; 5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"c-arithmetic-operations-with-integers\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-arithmetic-operations-with-integers\" class=\"header-mark\" aria-label=\"Direct link to section: C. Arithmetic Operations with Integers\"\u003e\u003c/a\u003eC. Arithmetic Operations with Integers\u003c/h4\u003e\u003ch5 id=\"1-addition\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-addition\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Addition\"\u003e\u003c/a\u003e1. Addition\u003c/h5\u003e\u003cp\u003eProperties of addition:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eClosure\u003c/strong\u003e: the sum of two integers is always an integer.\u003cbr\u003e\nExample: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-8 + 5 = -3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCommutative\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea + b = b + a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAssociative\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a + b) + c = a + (b + c)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eIdentity element\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea + 0 = 0 + a = a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eInverse element\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea + (-a) = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch5 id=\"2-subtraction\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-subtraction\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Subtraction\"\u003e\u003c/a\u003e2. Subtraction\u003c/h5\u003e\u003cp\u003eProperties of subtraction:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eClosure\u003c/strong\u003e: the difference of two integers is still an integer.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNot commutative\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea - b \\ne b - a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNot associative\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a - b) - c \\ne a - (b - c)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eThe following identities are useful:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na - b \u0026amp;= a + (-b)\\\\\na - (-b) \u0026amp;= a + b\\\\\n-a - b \u0026amp;= -(a+b)\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch5 id=\"3-multiplication\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-multiplication\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Multiplication\"\u003e\u003c/a\u003e3. Multiplication\u003c/h5\u003e\u003cp\u003eProperties of multiplication:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eClosure\u003c/strong\u003e: the product of two integers is always an integer.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCommutative\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\times b = b \\times a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eAssociative\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a \\times b) \\times c = a \\times (b \\times c)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eIdentity element\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\times 1 = 1 \\times a = a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDistributive\u003c/strong\u003e:\u003c/li\u003e\n\u003c/ul\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na \\times (b + c) \u0026amp;= ab + ac\\\\\na \\times (b - c) \u0026amp;= ab - ac\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eab\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eab\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch5 id=\"4-division\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-division\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Division\"\u003e\u003c/a\u003e4. Division\u003c/h5\u003e\u003cp\u003eProperties of division:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eNot closed\u003c/strong\u003e: the quotient of two integers is not always an integer.\u003cbr\u003e\nExample: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{12}{8} = \\dfrac{3}{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNot commutative\u003c/strong\u003e: \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{a}{b} \\ne \\dfrac{b}{a}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eNot associative\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003eDivision by zero is undefined.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"d-mixed-operations\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-mixed-operations\" class=\"header-mark\" aria-label=\"Direct link to section: D. Mixed Operations\"\u003e\u003c/a\u003eD. Mixed Operations\u003c/h4\u003e\u003cp\u003eUse these rules when working with mixed operations:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eEvaluate expressions inside parentheses first.\u003c/li\u003e\n\u003cli\u003eMultiplication and division are done before addition and subtraction.\u003c/li\u003e\n\u003cli\u003eFor operations at the same level, work from left to right.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"e-worked-examples\" class=\"headerLink\"\u003e\n \u003ca href=\"#e-worked-examples\" class=\"header-mark\" aria-label=\"Direct link to section: E. Worked Examples\"\u003e\u003c/a\u003eE. Worked Examples\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eFind the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-15 + (-12 : 3)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e19\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n -15 + (-12 : 3) = -15 + (-4) = -19\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e19\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eFind the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e18\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-18 + 30) : (-3 - 1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n 12 : (-4) = -3\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=-10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=-2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, find \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea + c - 3b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e34\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n a + c - 3b 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style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan 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class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e34\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e36\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep=-36\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eq=-9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er=8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, find the value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-p : q - r\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n \\begin{aligned}\n -p : q - r \u0026amp;= -(-36) : (-9) - 8\\\\\n \u0026amp;= 36 : (-9) - 8\\\\\n \u0026amp;= -4 - 8\\\\\n \u0026amp;= -12\n \\end{aligned}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThe classroom temperature is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e32\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e32^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eC. The office temperature is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e11^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eC lower. Therefore, the office temperature is\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e32\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmtext\u003eC\u003c/mtext\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmtext\u003eC\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmtext\u003eC\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n 32^\\circ\\text{C} - 11^\\circ\\text{C} = 21^\\circ\\text{C}\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8074em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eA pulley is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e meters above the ground, while the water surface is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e meters below the ground. The rope length from the water surface to the pulley is\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n 2 - (-6) = 2 + 6 = 8\n \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the rope is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e meters long.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"f-geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#f-geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: F. GeoGebra Simulation\"\u003e\u003c/a\u003eF. GeoGebra Simulation\u003c/h4\u003e\u003cp\u003eThe following GeoGebra simulation can be used to explore addition with integers.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/nukq7qsv/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Bilangan Bulat\" width=\"650px\"\u003e\n \u003c/iframe\u003e\n\u003ch4 id=\"g-practice-questions\" class=\"headerLink\"\u003e\n \u003ca href=\"#g-practice-questions\" class=\"header-mark\" aria-label=\"Direct link to section: G. Practice Questions\"\u003e\u003c/a\u003eG. Practice Questions\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eThe value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1 - 2 + 3 - 4 + 5 - 6 + \\cdots + 9 - 10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eA. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eB. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eC. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eD. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThe value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e that satisfies \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e15 : (p + 1) = -3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eA. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eB. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eC. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eD. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIn a test, each correct answer is worth \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, each wrong answer is worth \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and each unanswered question is worth \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Out of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e25\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e questions, Andika answered \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e18\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e18\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e correctly, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e incorrectly, and left the rest blank. Andika\u0026rsquo;s score is\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eA. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e60\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eB. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e61\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e61\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e61\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eC. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e62\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e62\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e62\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eD. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e63\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e63\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e63\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThe result of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e17\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5 \\times (17 + (-13))\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e17\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mclose\"\u003e))\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eA. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e98\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e98\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e98\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eB. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e91\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e91\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e91\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eC. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e85\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e85\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e85\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eD. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e72\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e72\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e72\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIf the initial temperature of an object is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e degrees and then it drops by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e degrees, the final temperature is\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eA. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eB. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eC. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eD. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eAnswer key\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eA, because \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(1-2)+(3-4)+(5-6)+(7-8)+(9-10)=-5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eB, because \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e15:(p+1)=-3 \\Rightarrow p+1=-5 \\Rightarrow p=-6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eC, because \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e18\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e72\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e62\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(18 \\times 4) + (5 \\times -2) = 72 - 10 = 62\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e72\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e62\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eD, because \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e17\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e17 + (-13) = 4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e17\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, so the result is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e20\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eD, because \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5 - 12 = -7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eIf you still find this topic difficult, review the examples above and discuss them with your teacher or classmates.\u003c/p\u003e\n\u003ch4 id=\"h-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#h-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: H. Evaluation\"\u003e\u003c/a\u003eH. Evaluation\u003c/h4\u003e\u003cp\u003eInteractive exercises for this lesson are available in the original source. They are not reproduced here because the imported legacy HTML and JavaScript are no longer stable in the current site.\u003c/p\u003e\n\u003ch4 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h4\u003e\u003cp\u003eThat completes this short introduction to integers. Once you understand the number line and the basic operation rules, integer problems become much easier to solve.\u003c/p\u003e\n", "language": "en" }, { "title" : "Driving School Program Training", "date_published" : "2021-06-07T15:19:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pelatihan-program-sekolah-penggerak/", "url" : "https://www.aantriono.com/en/pelatihan-program-sekolah-penggerak/", "summary": "On this occasion, I will tell you a little about the Driving School Program training that I have just participated in. What is the Driving School Program? The Driving School Program is a program to encourage the transformation process of educational units in order to improve student learning outcomes holistic both from the aspect of cognitive competence (literacy and numeracy) or non-cognitive (character) to realize the Pancasila Student Profile. Driving School Program is a program to encourage the transformation process of educational units so that they can improve student learning outcomes holistically from both aspects cognitive competence (literacy and numeracy) or non-cognitive (character) to realize the Pancasila Student Profile. Decree of the Minister of Education and Culture Republic of Indonesia Number 1177/M/2020 concerning Guidelines for Implementing the Driving School Program, states that the aim of the Driving School Program is to improve competence and character in accordance with the Pancasila Student Profile, guarantee equal distribution of educational quality through capacity building programs for school principals who are able to lead educational units in achieving quality learning, building a stronger educational ecosystem that focuses on improving quality, as well as creating a collaborative climate for stakeholders in the education sector, both at school and government levels regional and central. To achieve this goal, intervention at the educational unit level is carried out by strengthening school human resources through training and mentoring. Training and mentoring to strengthen school human resources uses a learning platform in the form of Learning Management System Driving School Program (LMS PSP) and facilitated by the instructor.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003eOn this occasion, I will tell you a little about the Driving School Program training that I have just participated in. What is the Driving School Program? The Driving School Program is a program to encourage the transformation process of educational units in order to improve student learning outcomes holistic both from the aspect of cognitive competence (literacy and numeracy) or non-cognitive (character) to realize the Pancasila Student Profile. Driving School Program is a program to encourage the transformation process of educational units so that they can improve student learning outcomes holistically from both aspects cognitive competence (literacy and numeracy) or non-cognitive (character) to realize the Pancasila Student Profile.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhRLThkSv5u0TUrLrYe-v7KtJ1hXMn07TR-cMZgWgTnGZp8iD-fM72ZYub9bUSPSw4kHJMNn3w_91wr28Qp_bIruP71vi5D3lKstGC1D5nDXmdvXicKjsD4rdDhAB8jytsv1kS_26PqRY/s650/sekolah-penggerak-ppt2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"pelatihanprogram sekolah penggerak\" border=\"0\" data-original-height=\"382\" data-original-width=\"650\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhRLThkSv5u0TUrLrYe-v7KtJ1hXMn07TR-cMZgWgTnGZp8iD-fM72ZYub9bUSPSw4kHJMNn3w_91wr28Qp_bIruP71vi5D3lKstGC1D5nDXmdvXicKjsD4rdDhAB8jytsv1kS_26PqRY/s16000/sekolah-penggerak-ppt2.png\" title=\"program sekolah penggerak\" /\u003e\u003c/a\u003e\nDecree of the Minister of Education and Culture Republic of Indonesia Number 1177/M/2020 concerning Guidelines for Implementing the Driving School Program, states that the aim of the Driving School Program is to improve competence and character in accordance with the Pancasila Student Profile, guarantee equal distribution of educational quality through capacity building programs for school principals who are able to lead educational units in achieving quality learning, building a stronger educational ecosystem that focuses on improving quality, as well as creating a collaborative climate for stakeholders in the education sector, both at school and government levels regional and central. To achieve this goal, intervention at the educational unit level is carried out by strengthening school human resources through training and mentoring. Training and mentoring to strengthen school human resources uses a learning platform in the form of Learning Management System Driving School Program (LMS PSP) and facilitated by the instructor.\u003c/p\u003e\n\u003cdiv id=\"toc\"\u003e\n\u003ch4 id=\"mobilization-school-program-training-participants\" class=\"headerLink\"\u003e\n \u003ca href=\"#mobilization-school-program-training-participants\" class=\"header-mark\" aria-label=\"Direct link to section: Mobilization School Program Training Participants\"\u003e\u003c/a\u003eMobilization School Program Training Participants\u003c/h4\u003e\u003cp\u003eAlhamdulillah, from 25 May - 7 June 2021 I had the opportunity to take part in the Phase 1 Driving School Program Training organized by \u003ca href=\"http://lppksps.kemdikbud.go.id/\" target=\"_blank\"\u003e\u003cstrong\u003eLPPKSPS\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e, with the instructor Mrs. Leny Tri Hartiastuti, S.Pd, and for the LMS I, Mr. Eko Rumekso, S.Kom. The participants are as follows:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eYustina Tresiah, S.Pd from SMP Negeri 3 Paritiga One Roof\u003c/li\u003e\n\u003cli\u003eUhti Amaliah, S.Pd from SMP Negeri 1 Simpang Teritip\u003c/li\u003e\n\u003cli\u003eDian Maulia, S.Pd from SMP Negeri 1 Muntok\u003c/li\u003e\n\u003cli\u003eRuliana Sofia, S.Pd from SMP Negeri 4 Metro\u003c/li\u003e\n\u003cli\u003eMufidatun Fajari, S.Pd from Muhammadiyah Ahmad Dahlan Middle School\u003c/li\u003e\n\u003cli\u003eMaya Pradipta Maharani, S.Pd from SMP Negeri 4 Krui\u003c/li\u003e\n\u003cli\u003eSyahrial, S.Pd from SMP Negeri 16 Krui\u003c/li\u003e\n\u003cli\u003eHeri Mulyanto, S.Pd from SMP Negeri 2 Punggur\u003c/li\u003e\n\u003cli\u003eAan Triono, S.Pd from Purworejo State Middle School\u003c/li\u003e\n\u003cli\u003eDian Puspita Dewi, S.Pd from SMP Negeri 1 Seputih Surabaya\u003c/li\u003e\n\u003cli\u003eWayan Putrayasa, S.Pd from SMP Negeri 3 Trimurjo\u003c/li\u003e\n\u003cli\u003eMurkanti Slamet, S.Pd from SMP Negeri 2 Gunung Sugih\u003c/li\u003e\n\u003cli\u003eMariya Yemi, S.Pd from SMP Negeri 2 Bangunrejo\u003c/li\u003e\n\u003cli\u003eHeri Sutrisno, S.Pd from SMP IT Bustanul Ulum\u003c/li\u003e\n\u003cli\u003eAnggi Yani Martinus Sari, S.Pd from SMP Negeri 2 Trimurjo\u003c/li\u003e\n\u003cli\u003eSunaryo, SE from SMP Negeri 1 Bandar Mataram\u003c/li\u003e\n\u003cli\u003eIskandar, S.Pd from SMP Negeri 1 Trimurjo\u003c/li\u003e\n\u003cli\u003eKasiati, S.Pd from SMP Negeri 1 Kalirejo\u003c/li\u003e\n\u003cli\u003eSri Sundari, S.Pd from SMP Negeri 1 Seputih Agung\u003c/li\u003e\n\u003cli\u003eSukamto, A.Md from SMP Negeri 2 Seputih Mataram\u003c/li\u003e\n\u003cli\u003eYurita Eka Sari, S.Pd from SMP Negeri 2 Selagai Lingga\u003c/li\u003e\n\u003cli\u003eSefilius Yulianti from SMP Kristen 2 bandar Jaya\u003c/li\u003e\n\u003cli\u003eShinta Shaputri, S.Pd from Cintailah IT Middle School\u003c/li\u003e\n\u003cli\u003eIsmiatin, S.Pd from SMP Islam Plus At Tholibin\u003c/li\u003e\n\u003cli\u003eRiska Apriani, S.Si from Muhammadiyah Seputih Middle School, Surabaya\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"driving-school-program-training-schedule\" class=\"headerLink\"\u003e\n \u003ca href=\"#driving-school-program-training-schedule\" class=\"header-mark\" aria-label=\"Direct link to section: Driving School Program Training Schedule\"\u003e\u003c/a\u003eDriving School Program Training Schedule\u003c/h4\u003e\u003cpre\u003e\u003ccode\u003e For the training schedule itself, there are two sessions, namely the morning session for participants from eastern Indonesia and the afternoon session for participants from western Indonesia. For more details, see the following attachment.\n\u003c/code\u003e\u003c/pre\u003e\n\u003ciframe height=\"480\" src=\"https://drive.google.com/file/d/1AN1A9y6pGKZrhr59241LYkGgmeof5loR/preview\" width=\"640\"\u003e\u003c/iframe\u003e\n\u003ch4 id=\"learning-patterns\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-patterns\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Patterns\"\u003e\u003c/a\u003eLearning Patterns\u003c/h4\u003e\u003cpre\u003e\u003ccode\u003e The learning pattern applied in the Driving School Program training activities is daily learning, where participants will be presented with a series of activities that must be carried out every day.\n The following is an overview of the learning patterns presented.\n \u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDz3yy1e6xs5bzbHlGnNpHCVcEbwaNIMG9j7cVZy9DOERuwS-W8a4pigIbUqWcvbgBsuO44hCqOpdcBUD8Djv9SmEpU3ZpSkZGf-1vjsw_noqUJaNSdcojQwaHNVUlw2D68al_a2HDBvQ/s1250/psp1.png\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\u0026lt;img alt=\u0026quot;LMS pelatihan Program Sekolah Penggerak\u0026quot; border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;588\u0026quot; data-original-width=\u0026quot;1250\u0026quot; height=\u0026quot;151\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDz3yy1e6xs5bzbHlGnNpHCVcEbwaNIMG9j7cVZy9DOERuwS-W8a4pigIbUqWcvbgBsuO44hCqOpdcBUD8Djv9SmEpU3ZpSkZGf-1vjsw_noqUJaNSdcojQwaHNVUlw2D68al_a2HDBvQ/w320-h151/psp1.png\u0026quot; title=\u0026quot;LMS pelatihan Program Sekolah Penggerak\u0026quot; width=\u0026quot;320\u0026quot; /\u0026gt;\u0026lt;/a\u0026gt;\n The numbers presented represent the number of days of learning, from day 1 to day 1. 10th day. The activities carried out will be different every day according to the training structure. The activities that will be carried out every day consist of activities *synchronous* And *asynchronous*. Activity *synchronous* is an activity to elaborate understanding with the instructor by providing reinforcement of understanding regarding the substance of the material, confirmation of learning, and reflection on learning obtained in independent asynchronous activities. Activity *synchronous* carried out in virtual classes via web meetings. The web meeting application used is Google Meet.\n \u0026lt;a href=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkgBLj2DjepnKpVqE5bkkvJlTiEHPqC4JPnNNUo0jH8x05IyQ7OGRRXHe2Hwyax1yeOim7IFSx3P1E71ScsF7wiRt4omM3EP9GgTY3CGnvH8yn_Wfp084af-fMOOc7YPXNbZZLYJqpyIc/s1280/WhatsApp+Image+2021-06-04+at+15.12.39.jpeg\u0026quot; style=\u0026quot;margin-left: 1em; margin-right: 1em;\u0026quot;\u0026gt;\u0026lt;img alt=\u0026quot;aktivitas synchronous di LMS PSP\u0026quot; border=\u0026quot;0\u0026quot; data-original-height=\u0026quot;719\u0026quot; data-original-width=\u0026quot;1280\u0026quot; height=\u0026quot;180\u0026quot; src=\u0026quot;https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkgBLj2DjepnKpVqE5bkkvJlTiEHPqC4JPnNNUo0jH8x05IyQ7OGRRXHe2Hwyax1yeOim7IFSx3P1E71ScsF7wiRt4omM3EP9GgTY3CGnvH8yn_Wfp084af-fMOOc7YPXNbZZLYJqpyIc/w320-h180/WhatsApp+Image+2021-06-04+at+15.12.39.jpeg\u0026quot; title=\u0026quot;aktivitas synchronous di LMS\u0026quot; width=\u0026quot;320\u0026quot; /\u0026gt;\u0026lt;/a\u0026gt;\n Meanwhile on activities *asynchronous* participants read the material, watch learning videos before they come to elaborate their understanding with the facilitator in the virtual class and they start to discuss, exchange knowledge, solve problems with other participants, activities to answer reflective questions, activities to study material and discuss material with other participants, collaborative activities in groups with other participants, activities to do independent assignments and upload assignments, reflection activities on blogs, and the activity of answering questions on the quiz. All activities *asynchronous* done within the LMS. I have implemented Moodle LMS to carry out distance learning during the Covid-19 pandemic. Please click on the E-LEARNING menu on this blog or click on **\u0026lt;a href=\u0026quot;https://aantriono.gnomio.com/login/index.php\u0026quot; target=\u0026quot;_blank\u0026quot;\u0026gt;here\u0026lt;/a\u0026gt;\n\u003c/code\u003e\u003c/pre\u003e\n\u003cp\u003e. **\n**\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCEuF-ulMQl4Ngu-0ftzV2qjUVE24jtp1g8NCs-brkJCHaXsUOBvhYqHpZuhvpmps-OCDk-zu_dB4g-H5CpbcoA60Gl6q3hIXAa2FOEqrYW3hrj52OmM6eXByxx4i_FWlPXHhI0KWYWOg/s1280/WhatsApp+Image+2021-06-08+at+07.40.17.jpeg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"aktivitas asynchronous di LMS PSP\" border=\"0\" data-original-height=\"641\" data-original-width=\"1280\" height=\"160\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCEuF-ulMQl4Ngu-0ftzV2qjUVE24jtp1g8NCs-brkJCHaXsUOBvhYqHpZuhvpmps-OCDk-zu_dB4g-H5CpbcoA60Gl6q3hIXAa2FOEqrYW3hrj52OmM6eXByxx4i_FWlPXHhI0KWYWOg/w320-h160/WhatsApp+Image+2021-06-08+at+07.40.17.jpeg\" title=\"aktivitas asynchronous di LMS PSP\" width=\"320\" /\u003e\u003c/a\u003e\n**\u003c/p\u003e\n\u003ch4 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h4\u003e\u003cp\u003eThe material studied in this training is:\u003c/p\u003e\n\u003col\u003e\n \u003cli\u003eDriving School Program Policy\u003c/li\u003e\n \u003cli\u003eTraining orientation\u003c/li\u003e\n \u003cli\u003eLiberating education\u003c/li\u003e\n \u003cli\u003eCurriculum framework\u003c/li\u003e\n \u003cli\u003eLearning outcomes\u003c/li\u003e\n \u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/08/mengenal-platform-merdeka-mengajar_3.html\" target=\"_blank\"\u003eTeaching tools\u003c/a\u003e\n\u003c/li\u003e\n \u003cli\u003e\u003ca href=\"https://www.aantriono.com/2021/08/mengenal-platform-merdeka-mengajar_3.html\" target=\"_blank\"\u003eAssessment\u003c/a\u003e\n\u003c/li\u003e\n \u003cli\u003eOperational curriculum in educational units\u003c/li\u003e\n \u003cli\u003eIntroduction to digital literacy\u003c/li\u003e\n \u003cli\u003eImplementation of IHT in educational units\u003c/li\u003e\n \u003c/ol\u003e\n For a summary of the driving school program training materials, please click the following button.\n \u003ca class=\"button\" href=\"https://drive.google.com/file/d/1yuuMwO_p2Avd6U7qVFO2iyGe-UsUWWR9/view?usp=sharing\"\u003eDownloads\u003c/a\u003e\n\u003ch4 id=\"constraint\" class=\"headerLink\"\u003e\n \u003ca href=\"#constraint\" class=\"header-mark\" aria-label=\"Direct link to section: Constraint\"\u003e\u003c/a\u003eConstraint\u003c/h4\u003e\u003cp\u003eBecause this training uses online mode, of course there are several obstacles experienced by the participants, such as not being familiar with the Moodle LMS used, the internet network being unstable so that it affects when participating in activities. \u003cem\u003esynchronous\u003c/em\u003e in Google Meet or in activities \u003cem\u003easynchronous\u003c/em\u003e working on and uploading assignments in the LMS, apart from that there is not much time to complete the assignments, so it seems like you are pressed for time so that all assignments are checked by the LMS system. Not to mention the health problems that are disrupted due to lack of rest during this training.\u003c/p\u003e\n\u003ch4 id=\"benefit\" class=\"headerLink\"\u003e\n \u003ca href=\"#benefit\" class=\"header-mark\" aria-label=\"Direct link to section: Benefit\"\u003e\u003c/a\u003eBenefit\u003c/h4\u003e\u003cp\u003eOf course, schools that take part in the Driving School Program will gain many benefits, including:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eIncreasing the competence of school principals and teachers\u003c/li\u003e\n \u003cli\u003eImproving the quality of learning outcomes within a period of 3 years\u003c/li\u003e\n \u003cli\u003eAccelerating school digitalization\u003c/li\u003e\n \u003cli\u003eAccelerating the achievement of the Pancasila Student Profile\u003c/li\u003e\n \u003cli\u003eOpportunities to become a catalyst for change for educational units\u003c/li\u003e\n \u003cli\u003eGet intensive assistance\u003c/li\u003e\n \u003cli\u003eObtain additional budget to purchase books for learning with a new paradigm\u003c/li\u003e\n \u003c/ul\u003e\n\u003ch4 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h4\u003e\u003cpre\u003e\u003ccode\u003e Can't believe today, Monday 07 June 2021 is our last day taking part in the Driving School Program training. There are many impressions and hopes from all of us regarding this training and its implementation in our respective educational units entering the 2021/2022 academic year. Thanks and high appreciation are expressed to Mrs. Leny Tri Hartiastuti, S.Pd and Mr. Eko Rumekso, S.Kom for all their knowledge, guidance and enlightenment so that we understand about driving schools. I, don't forget to thank all the participating fathers/mothers for all the good cooperation, for the willingness to share and collaborate so far, hopefully all your good deeds will be accepted by Allah Subhanahu wa Ta'ala. And May Allah Almighty always bless the efforts we make to achieve Indonesia's educational goals. Amen.\n\u003c/code\u003e\u003c/pre\u003e\n\u003c/div\u003e\n That's the article about the driving school program training. If you have suggestions and input, please leave them in the comments. I hope this is helpful.\n", "language": "en" }, { "title" : "Probability", "date_published" : "2021-03-08T06:38:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/peluang/", "url" : "https://www.aantriono.com/en/peluang/", "summary": " Basics of ProbabilityIn daily life, we often hear statements such as:\nIt will probably rain this afternoon. Joni has a big chance of becoming class president. Indonesia has only a small chance of beating Brazil in a football match. Words like probably, big chance, and small chance describe how likely an event is. In mathematics, that likelihood can be studied more precisely.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"basics-of-probability\" class=\"headerLink\"\u003e\n \u003ca href=\"#basics-of-probability\" class=\"header-mark\" aria-label=\"Direct link to section: Basics of Probability\"\u003e\u003c/a\u003eBasics of Probability\u003c/h2\u003e\u003cp\u003eIn daily life, we often hear statements such as:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eIt will probably rain this afternoon.\u003c/li\u003e\n\u003cli\u003eJoni has a big chance of becoming class president.\u003c/li\u003e\n\u003cli\u003eIndonesia has only a small chance of beating Brazil in a football match.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eWords like \u003cem\u003eprobably\u003c/em\u003e, \u003cem\u003ebig chance\u003c/em\u003e, and \u003cem\u003esmall chance\u003c/em\u003e describe how likely an event is. In mathematics, that likelihood can be studied more precisely.\u003c/p\u003e\n\u003ch2 id=\"random-events\" class=\"headerLink\"\u003e\n \u003ca href=\"#random-events\" class=\"header-mark\" aria-label=\"Direct link to section: Random Events\"\u003e\u003c/a\u003eRandom Events\u003c/h2\u003e\u003cp\u003eTry tossing a coin. We cannot know in advance whether the number side or the picture side will appear. This is an example of a \u003cstrong\u003erandom event\u003c/strong\u003e, an event whose outcome cannot be determined before the experiment is carried out.\u003c/p\u003e\n\u003cp\u003eSimple activity:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ePrepare one die, one container, five red balls, and five yellow balls.\u003c/li\u003e\n\u003cli\u003eRoll the die. Can you tell which face will appear?\u003c/li\u003e\n\u003cli\u003ePut all the balls into the container, mix them, close your eyes, and take one ball.\u003c/li\u003e\n\u003cli\u003eRepeat without closing your eyes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eFrom this activity, we see that some outcomes cannot be predicted beforehand. Those are random events.\u003c/p\u003e\n\u003ch2 id=\"sample-points-and-sample-space\" class=\"headerLink\"\u003e\n \u003ca href=\"#sample-points-and-sample-space\" class=\"header-mark\" aria-label=\"Direct link to section: Sample Points and Sample Space\"\u003e\u003c/a\u003eSample Points and Sample Space\u003c/h2\u003e\u003cp\u003eWhen a coin is tossed, the possible outcomes are the number side \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(A)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the picture side \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(G)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{A, G\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe set of all possible outcomes is called the \u003cstrong\u003esample space\u003c/strong\u003e, denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Its members are called \u003cstrong\u003esample points\u003c/strong\u003e. The number of members in the sample space is written as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(S)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThere are three common ways to determine a sample space:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eby listing,\u003c/li\u003e\n\u003cli\u003eby using a table,\u003c/li\u003e\n\u003cli\u003eby using a tree diagram.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"1-determining-the-sample-space-by-listing\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-determining-the-sample-space-by-listing\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Determining the Sample Space by Listing\"\u003e\u003c/a\u003e1. Determining the Sample Space by Listing\u003c/h3\u003e\u003cp\u003eIf two coins are tossed at the same time, the possible outcomes are:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{AA, AG, GA, GG\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eAA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eGG\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nn(S) = 4\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-determining-the-sample-space-with-a-table\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-determining-the-sample-space-with-a-table\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Determining the Sample Space with a Table\"\u003e\u003c/a\u003e2. Determining the Sample Space with a Table\u003c/h3\u003e\u003cp\u003eThe sample space for two coins can also be shown with a table:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eA\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eG\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eA\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAA\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eAG\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eG\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eGA\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eGG\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{AA, AG, GA, GG\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eAA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eGG\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nn(S) = 4\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-determining-the-sample-space-with-a-tree-diagram\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-determining-the-sample-space-with-a-tree-diagram\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Determining the Sample Space with a Tree Diagram\"\u003e\u003c/a\u003e3. Determining the Sample Space with a Tree Diagram\u003c/h3\u003e\u003cp\u003eAnother simple way is to use a tree diagram.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrLunVHrEVihbqucx3PgP_Ghytd3w4nX813hAuZVW0JkZOGtI6cM6dhSwe67fPs3aHUapCz2qoT4bG_pv_rmaEWPuztk_SAGgCRYYxgHxSrpps6XjKCJbN9-AhrdGfRxA9q5txrxPZ3qg/w320-h178/peluang1.png' alt=\"Tree diagram for two coins\" \u003e\u003c/p\u003e\n\u003cp\u003eThe result is still:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{AA, AG, GA, GG\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eAA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eGG\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nn(S) = 4\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-problem\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eDetermine the sample space for the following experiments:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003erolling one die,\u003c/li\u003e\n\u003cli\u003etossing three coins at once,\u003c/li\u003e\n\u003cli\u003erolling two dice at once.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eFor one die:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{1, 2, 3, 4, 5, 6\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eFor three coins, a tree diagram can be used.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDfXOy-QRhmJqoyLewelAhy6vu7qX5aR5CYlvaUjvPfyh_wL7Fg9B0x7EyheFgKnUBnY82LIAELPiS39ZQpgdtsxXITADDIyrMI6cr7OhDhqS25ZuyqNoApksaTm5H-Jqtgobb9iCwCXY/w320-h222/peluang2.png' alt=\"Tree diagram for three coins\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eAAA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eAA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eGG\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eAA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eGG\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eGGG\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"3\"\u003e\n\u003cli\u003eFor two dice, a table can be used.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMGniMd3SKi1ZCNmW9pcx-lJsoIwkp7KNTvwjxls-gxp_w8lVMjctVKptoZf9N7o4ZJ0b08ax4qUSirKVGpEbk39L4rGbiFVwN8RQ4Q935tBPZ23Vwd5rpTtLGekBEPvAKwS7WUVbUKnU/w400-h204/peluang3.png' alt=\"Table for two dice\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e…\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{(1,1), (1,2), (1,3), \\ldots, (6,6)\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e…\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"calculating-probability\" class=\"headerLink\"\u003e\n \u003ca href=\"#calculating-probability\" class=\"header-mark\" aria-label=\"Direct link to section: Calculating Probability\"\u003e\u003c/a\u003eCalculating Probability\u003c/h2\u003e\u003ch3 id=\"1-understanding-events\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-understanding-events\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Understanding Events\"\u003e\u003c/a\u003e1. Understanding Events\u003c/h3\u003e\u003cp\u003eIn rolling one die, the sample space is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{1, 2, 3, 4, 5, 6\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAny subset of the sample space is called an \u003cstrong\u003eevent\u003c/strong\u003e. For example:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nK = \\{2, 4, 6\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eis the event of getting an even number, so:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nn(K) = 3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-probability-using-relative-frequency\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-probability-using-relative-frequency\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Probability Using Relative Frequency\"\u003e\u003c/a\u003e2. Probability Using Relative Frequency\u003c/h3\u003e\u003cp\u003eRelative frequency is the ratio between the number of observed events and the number of trials.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eRelative frequency\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmtext\u003enumber of event \u003c/mtext\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cmtext\u003enumber of trials\u003c/mtext\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\text{Relative frequency} = \\frac{\\text{number of event }K}{\\text{number of trials}}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eRelative frequency\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003enumber of trials\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003enumber of event \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf a coin is tossed 30 times and the picture side appears 13 times, then the relative frequency is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmn\u003e30\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{13}{30}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-problem-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eRino rolls a die 200 times. The results are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eface 1 appears 25 times,\u003c/li\u003e\n\u003cli\u003eface 3 appears 17 times,\u003c/li\u003e\n\u003cli\u003eface 6 appears 56 times.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eFind the relative frequency of each event.\u003c/p\u003e\n\u003ch4 id=\"solution-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-1\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eFor face 1:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eRelative frequency\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmn\u003e200\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.125\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Relative frequency} \u0026amp;= \\frac{25}{200} \\\\\n\u0026amp;= \\frac{1}{8} \\\\\n\u0026amp;= 0.125\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.1149em;vertical-align:-2.8074em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.3074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eRelative frequency\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.3074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e200\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.125\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eFor face 3:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eRelative frequency\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e17\u003c/mn\u003e\u003cmn\u003e200\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.085\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Relative frequency} \u0026amp;= \\frac{17}{200} \\\\\n\u0026amp;= 0.085\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.8074em;vertical-align:-1.6537em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.1537em;\"\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eRelative frequency\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.6537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.1537em;\"\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e200\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e17\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.085\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.6537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"3\"\u003e\n\u003cli\u003eFor face 6:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eRelative frequency\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e56\u003c/mn\u003e\u003cmn\u003e200\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.28\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Relative frequency} \u0026amp;= \\frac{56}{200} \\\\\n\u0026amp;= 0.28\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.8074em;vertical-align:-1.6537em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.1537em;\"\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eRelative frequency\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.6537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.1537em;\"\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e200\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e56\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.28\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.6537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-probability-using-the-formula\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-probability-using-the-formula\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Probability Using the Formula\"\u003e\u003c/a\u003e3. Probability Using the Formula\u003c/h3\u003e\u003cp\u003eIf every sample point is equally likely, then the probability of event \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(K) = \\frac{n(K)}{n(S)}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.363em;vertical-align:-0.936em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.427em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFor example, when rolling a die:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{1, 2, 3, 4, 5, 6\\}, \\qquad n(S) = 6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf the event of prime numbers is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nK = \\{2, 3, 5\\}, \\qquad n(K)=3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(K) = \\frac{3}{6} = \\frac{1}{2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-problem-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eSiti throws one die. Determine the probability of getting:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ea 3,\u003c/li\u003e\n\u003cli\u003ea number greater than 3,\u003c/li\u003e\n\u003cli\u003eone of 1, 2, 3, 4, 5, 6,\u003c/li\u003e\n\u003cli\u003ea number greater than 6.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"solution-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-2\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003cp\u003eSince:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{1, 2, 3, 4, 5, 6\\}, \\qquad n(S)=6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA = \\{3\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(A) = \\frac{1}{6}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB = \\{4, 5, 6\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(B) = \\frac{3}{6} = \\frac{1}{2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"3\"\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC = \\{1, 2, 3, 4, 5, 6\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(C) = \\frac{6}{6} = 1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"4\"\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∅\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD = \\varnothing\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6633em;vertical-align:-0.0817em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord amsrm\"\u003e∅\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(D) = 0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"probability-values\" class=\"headerLink\"\u003e\n \u003ca href=\"#probability-values\" class=\"header-mark\" aria-label=\"Direct link to section: Probability Values\"\u003e\u003c/a\u003eProbability Values\u003c/h2\u003e\u003cp\u003eProbability values always satisfy:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n0 \\leq P(K) \\leq 1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7804em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThis means:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP(K)=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e means the event is impossible.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP(K)=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e means the event is certain.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0 \u0026lt; P(K) \u0026lt; 1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6835em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e means the event may happen.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the complement of event \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(L) = 1 - P(K)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(L) + P(K) = 1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eExample:\u003c/p\u003e\n\u003cp\u003eIf the probability that Romi passes an exam is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e0.9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0.9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then the probability that Romi does not pass is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e0.9\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n1 - 0.9 = 0.1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-problem-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eFifteen cards are numbered from 1 to 15. One card is drawn at random. Determine the probability of drawing:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ean even number,\u003c/li\u003e\n\u003cli\u003ea non-even number.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"solution-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-3\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003cp\u003eThe sample space is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nS = \\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col\u003e\n\u003cli\u003eEven numbers:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = \\{2, 4, 6, 8, 10, 12, 14\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eso:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(A) = \\frac{7}{15}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eNon-even numbers:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB = \\{1, 3, 5, 7, 9, 11, 13, 15\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eso:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(B) = \\frac{8}{15}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the complement of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(B) = 1 - \\frac{7}{15} = \\frac{8}{15}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"expected-frequency\" class=\"headerLink\"\u003e\n \u003ca href=\"#expected-frequency\" class=\"header-mark\" aria-label=\"Direct link to section: Expected Frequency\"\u003e\u003c/a\u003eExpected Frequency\u003c/h2\u003e\u003cp\u003eExpected frequency is the predicted number of times an event occurs in a certain number of trials.\u003c/p\u003e\n\u003cp\u003eIts formula is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nF_h = P(K) \\times n\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eF_h\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = expected frequency,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP(K)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = probability of event \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = number of trials.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"example-problem-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem-4\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eA coin is tossed 30 times. Determine the expected frequency of the number side appearing.\u003c/p\u003e\n\u003ch4 id=\"solution-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-4\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003cp\u003eSince:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(K)=\\frac{1}{2}, \\qquad n=30\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nF_h \u0026amp;= P(K)\\times n \\\\\n\u0026amp;= \\frac{1}{2}\\times 30 \\\\\n\u0026amp;= 15\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3074em;vertical-align:-2.4037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-problem-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem-5\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eA die is rolled 100 times. Determine the expected frequency of:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ea prime-numbered face,\u003c/li\u003e\n\u003cli\u003ea face numbered less than 3.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"solution-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-5\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003col\u003e\n\u003cli\u003ePrime numbers on a die:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA = \\{2, 3, 5\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eso:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(A)=\\frac{3}{6}=\\frac{1}{2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e50\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nF_h = \\frac{1}{2}\\times 100 = 50\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e50\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eNumbers less than 3:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB = \\{1, 2\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eso:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nP(B)=\\frac{2}{6}=\\frac{1}{3}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nF_h = \\frac{1}{3}\\times 100 = \\frac{100}{3}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-problem-6\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problem-6\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problem\"\u003e\u003c/a\u003eExample Problem\u003c/h3\u003e\u003cp\u003eIn one area, the probability that a child contracts a disease is 0.05. Determine how many children are expected to contract the disease in a sample of 1,000 children.\u003c/p\u003e\n\u003ch4 id=\"solution-6\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-6\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.05\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e1,000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e50\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nF_h \u0026amp;= P(K)\\times n \\\\\n\u0026amp;= 0.05 \\times 1{,}000 \\\\\n\u0026amp;= 50\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.05\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e50\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the expected number of children who contract the disease is 50.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat completes this lesson on probability. Corrections and suggestions can be shared through the available feedback channel.\u003c/p\u003e\n", "language": "en" }, { "title" : "Statistics", "date_published" : "2021-02-20T14:36:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/02/statistika/", "url" : "https://www.aantriono.com/2021/02/statistika/", "summary": " Introduction A. Basic Competencies3.10 Analyze data based on data distribution, mean, median, and mode in order to draw conclusions, make decisions, and make predictions.\n4.10 Present and solve problems related to data distribution, mean, median, mode, and data spread in order to draw conclusions, make decisions, and make predictions.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch3 id=\"a-basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: A. Basic Competencies\"\u003e\u003c/a\u003eA. Basic Competencies\u003c/h3\u003e\u003cp\u003e3.10 Analyze data based on data distribution, mean, median, and mode in order to draw conclusions, make decisions, and make predictions.\u003c/p\u003e\n\u003cp\u003e4.10 Present and solve problems related to data distribution, mean, median, mode, and data spread in order to draw conclusions, make decisions, and make predictions.\u003c/p\u003e\n\u003ch3 id=\"b-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: B. Learning Objectives\"\u003e\u003c/a\u003eB. Learning Objectives\u003c/h3\u003e\u003cp\u003eAfter studying statistics, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eanalyze data using distribution, mean, median, and mode,\u003c/li\u003e\n\u003cli\u003epresent data in several forms,\u003c/li\u003e\n\u003cli\u003esolve problems related to data presentation and interpretation.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-material\" class=\"header-mark\" aria-label=\"Direct link to section: C. Material\"\u003e\u003c/a\u003eC. Material\u003c/h2\u003e\u003ch2 id=\"data-presentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#data-presentation\" class=\"header-mark\" aria-label=\"Direct link to section: Data Presentation\"\u003e\u003c/a\u003eData Presentation\u003c/h2\u003e\u003ch3 id=\"1-understanding-data-and-statistics\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-understanding-data-and-statistics\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Understanding Data and Statistics\"\u003e\u003c/a\u003e1. Understanding Data and Statistics\u003c/h3\u003e\u003cp\u003eData is a collection of datums, and a datum is a single fact.\u003c/p\u003e\n\u003cp\u003eExample: Ratna measures the heights of five students and obtains the following results.\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eName\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eDwi\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eWili\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eNita\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eWulan\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eDani\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eHeight (cm)\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e155\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e160\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e158\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e160\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e165\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eThe number \u003cstrong\u003e155 cm\u003c/strong\u003e is one datum. The complete set of results is called \u003cstrong\u003edata\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eFrom the table above, we can conclude that:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eDani is the tallest,\u003c/li\u003e\n\u003cli\u003eDwi is the shortest,\u003c/li\u003e\n\u003cli\u003eWili and Wulan have the same height.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eDrawing conclusions from data is part of \u003cstrong\u003estatistics\u003c/strong\u003e, the science of collecting, processing, and interpreting data.\u003c/p\u003e\n\u003cp\u003eThere are two general types of data:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eQuantitative data\u003c/strong\u003e, which is numerical,\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eQualitative data\u003c/strong\u003e, which describes characteristics or conditions.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"2-population-and-sample\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-population-and-sample\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Population and Sample\"\u003e\u003c/a\u003e2. Population and Sample\u003c/h3\u003e\u003cp\u003eIn many cases, conclusions are not based on all available data. For example, to test river water quality, a researcher does not need to test all the water in the river.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eall the water in the river is the \u003cstrong\u003epopulation\u003c/strong\u003e,\u003c/li\u003e\n\u003cli\u003ethe water taken for testing is the \u003cstrong\u003esample\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"3-presenting-data-in-tables\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-presenting-data-in-tables\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Presenting Data in Tables\"\u003e\u003c/a\u003e3. Presenting Data in Tables\u003c/h3\u003e\u003cp\u003eThe mathematics test scores of 30 students are:\u003c/p\u003e\n\u003cp\u003e6, 8, 7, 6, 6, 5, 7, 8, 8, 5,\u003cbr\u003e\n9, 9, 8, 6, 7, 7, 7, 6, 8, 7,\u003cbr\u003e\n10, 8, 8, 6, 6, 5, 9, 9, 7, 6\u003c/p\u003e\n\u003cp\u003eIn a frequency table:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: right\"\u003eScore\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eTally\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eNumber of Students\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eIII\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e6\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eIIIII III\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e8\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e7\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eIIIII II\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e7\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e8\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eIIIII II\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e7\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e9\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eIIII\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eI\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e1\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003eTotal\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e30\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"presenting-data-in-diagrams\" class=\"headerLink\"\u003e\n \u003ca href=\"#presenting-data-in-diagrams\" class=\"header-mark\" aria-label=\"Direct link to section: Presenting Data in Diagrams\"\u003e\u003c/a\u003ePresenting Data in Diagrams\u003c/h2\u003e\u003ch3 id=\"a-pictogram\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-pictogram\" class=\"header-mark\" aria-label=\"Direct link to section: a. Pictogram\"\u003e\u003c/a\u003ea. Pictogram\u003c/h3\u003e\u003cp\u003eA pictogram uses pictures to represent data.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ePopulation data:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eArea A = 800 people\u003c/li\u003e\n\u003cli\u003eArea B = 650 people\u003c/li\u003e\n\u003cli\u003eArea C = 700 people\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003ePictogram:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbPVkloEXuIINKiwdk1KSvWSTo4oIVriCfY9PsSlLFXeHME6XDdzTe4M1vF0f0btYSF6K27E-G8YgSjkgvaFn1oqZrcmP-Ux-UtXMrGerJTbfZHof_pcrNToxHw9mITOdGjxGTjRqJEpo/s482/statistik1.png' alt=\"Pictogram\" \u003e\u003c/p\u003e\n\u003ch3 id=\"b-bar-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-bar-chart\" class=\"header-mark\" aria-label=\"Direct link to section: b. Bar Chart\"\u003e\u003c/a\u003eb. Bar Chart\u003c/h3\u003e\u003cp\u003eBar charts are suitable for categorical data.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eCity\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eA\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eB\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eC\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eD\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eE\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eMaximum Temperature (°C)\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e12\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e20\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eMinimum Temperature (°C)\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e25\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e30\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e32\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e27\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e35\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eVertical bar chart:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjTBnEVv-msG3G4MhV9gjwdmvaQaQ8Zp00FM82L9QhBVyDuJzqSvavpTPhL4Yy4yg0LzMXuYckbMmNF4c8rIU27IKL5pQQwmKVfWts8X2PIiBh1CyIrNL0cXqP54iNWwXJyEqIKaJfbAw/s294/statistik2.png' alt=\"Vertical bar chart\" \u003e\u003c/p\u003e\n\u003cp\u003eHorizontal bar chart:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYjV-Ckw2ZVFfUUhlRv76r2RF0nUtdYl-MtkdmTfclaeSe1u-lm-ClAd-8R745sLf35w_qHB8sGf8CxOZf6UVcOym41ePqFl1KQ_9xUC7jPMzvCzt044VYMCT3BIz3QTAoVqQknGKC-zk/s385/statistik3.png' alt=\"Horizontal bar chart\" \u003e\u003c/p\u003e\n\u003ch3 id=\"c-line-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-line-chart\" class=\"header-mark\" aria-label=\"Direct link to section: c. Line Chart\"\u003e\u003c/a\u003ec. Line Chart\u003c/h3\u003e\u003cp\u003eLine charts are useful for continuous or periodic data.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eMonth\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eJan\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eFeb\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eMar\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eApr\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eMay\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eJun\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eJul\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eAug\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eSep\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eOct\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eNov\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eDec\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eNumber of TVs\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e20\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e12\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e17\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e20\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e25\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eLine chart:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8LfQGO0tG8rtQxJhYZ0vSkYvELr_UZMmo1AqVa17EK6JPu9ZU71Q81gsPbY12EGHQQkK-VCv_M5rZMFEqVGhHNutVRIqMbmVg7RJtozmew5EfGrEaBJCtOn5Z8huAvDyUig6dxp1dJBk/s619/statistik4.png' alt=\"Line chart\" \u003e\u003c/p\u003e\n\u003ch3 id=\"d-pie-chart\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-pie-chart\" class=\"header-mark\" aria-label=\"Direct link to section: d. Pie Chart\"\u003e\u003c/a\u003ed. Pie Chart\u003c/h3\u003e\u003cp\u003ePie charts show the comparison between parts and the whole.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eColor\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eFrequency\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eWhite\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003ePink\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eRed\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e8\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eBlue\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e8\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eYellow\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003eGreen\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eCentral angles:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eWhite\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003ePink\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eRed\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e72\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eBlue\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e72\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eYellow\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e45\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eGreen\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e45\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{White}\u0026amp;=\\frac{10}{40}\\times 360^\\circ=90^\\circ\\\\\n\\text{Pink}\u0026amp;=\\frac{4}{40}\\times 360^\\circ=36^\\circ\\\\\n\\text{Red}\u0026amp;=\\frac{8}{40}\\times 360^\\circ=72^\\circ\\\\\n\\text{Blue}\u0026amp;=\\frac{8}{40}\\times 360^\\circ=72^\\circ\\\\\n\\text{Yellow}\u0026amp;=\\frac{5}{40}\\times 360^\\circ=45^\\circ\\\\\n\\text{Green}\u0026amp;=\\frac{5}{40}\\times 360^\\circ=45^\\circ\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:13.8446em;vertical-align:-6.6723em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:7.1723em;\"\u003e\u003cspan style=\"top:-9.1723em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eWhite\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.8649em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ePink\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eRed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eBlue\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eYellow\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:2.3649em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eGreen\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.6723em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:7.1723em;\"\u003e\u003cspan style=\"top:-9.1723em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.8649em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:2.3649em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.6723em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ePie chart:\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEha8_bPX3n_kx9FRc8ANpLqwn7ZMzwn_mDzfTbaXAxwzTGkKwY__fkRidlZdgHrBYbfJ22M8hAWFVB2mKK0XewOOz5NN_IBAczGgI7yIPIg4a2mhOjDirjuO1IPVP1FwFXsVYFjJChN8IE/s202/statistik5.png' alt=\"Pie chart\" \u003e\u003c/p\u003e\n\u003ch2 id=\"b-measures-of-central-tendency\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-measures-of-central-tendency\" class=\"header-mark\" aria-label=\"Direct link to section: B. Measures of Central Tendency\"\u003e\u003c/a\u003eB. Measures of Central Tendency\u003c/h2\u003e\u003ch3 id=\"1-mean\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-mean\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Mean\"\u003e\u003c/a\u003e1. Mean\u003c/h3\u003e\u003cp\u003eThe mean is the sum of all data values divided by the number of data values.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003eˉ\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\bar{x}=\\frac{x_1+x_2+\\cdots+x_n}{n}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5678em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord accent\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5678em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"accent-body\" style=\"left:-0.2222em;\"\u003e\u003cspan class=\"mord\"\u003eˉ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.9463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cstrong\u003eExample 1\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData: 8, 8, 6, 7, 6, 7, 9, 9\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003eˉ\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7.5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\bar{x}=\\frac{60}{8}=7.5\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5678em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord accent\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5678em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"accent-body\" style=\"left:-0.2222em;\"\u003e\u003cspan class=\"mord\"\u003eˉ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7.5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cstrong\u003eExample 2\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe average score of 10 students is 70. After adding Rino\u0026rsquo;s score, the average becomes 68.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e700\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e68\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e700\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e748\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e700\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e48\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx_1+x_2+\\cdots+x_{10}\u0026amp;=700\\\\\n68\u0026amp;=\\frac{700+x_{11}}{11}\\\\\n748\u0026amp;=700+x_{11}\\\\\nx_{11}\u0026amp;=48\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.8074em;vertical-align:-3.1537em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6537em;\"\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e68\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e748\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6537em;\"\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e700\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e700\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e700\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e48\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, Rino\u0026rsquo;s score is \u003cstrong\u003e48\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eFor frequency data:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003eˉ\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\bar{x}=\\frac{f_1x_1+f_2x_2+\\cdots+f_ix_i}{f_1+f_2+\\cdots+f_i}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5678em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord accent\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5678em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"accent-body\" style=\"left:-0.2222em;\"\u003e\u003cspan class=\"mord\"\u003eˉ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2519em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: right\"\u003eWeight \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_i)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eFrequency \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(f_i)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef_i x_i\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e42\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e84\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e43\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e129\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e44\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e44\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e45\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e180\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003eTotal\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e437\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003eˉ\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e437\u003c/mn\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e43.7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\bar{x}=\\frac{437}{10}=43.7\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5678em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord accent\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5678em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"accent-body\" style=\"left:-0.2222em;\"\u003e\u003cspan class=\"mord\"\u003eˉ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e437\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e43.7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-mode\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-mode\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Mode\"\u003e\u003c/a\u003e2. Mode\u003c/h3\u003e\u003cp\u003eThe mode is the value that appears most often in a data set.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData:\u003c/p\u003e\n\u003cp\u003e1, 4, 3, 5, 2, 3, 2, 2, 5, 4, 3, 1\u003c/p\u003e\n\u003cp\u003eThe most frequent value is \u003cstrong\u003e2\u003c/strong\u003e, so the mode is \u003cstrong\u003e2\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"3-median\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-median\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Median\"\u003e\u003c/a\u003e3. Median\u003c/h3\u003e\u003cp\u003eThe median is the middle value of ordered data.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eFor an odd number of data values, the median is the middle value.\u003c/li\u003e\n\u003cli\u003eFor an even number of data values, the median is the average of the two middle values.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cstrong\u003eExample 1\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData: 6, 7, 6, 6, 5, 8, 7\u003c/p\u003e\n\u003cp\u003eOrdered:\u003c/p\u003e\n\u003cp\u003e5, 6, 6, 6, 7, 7, 8\u003c/p\u003e\n\u003cp\u003eMedian = \u003cstrong\u003e6\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eExample 2\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData: 7, 7, 10, 8, 6, 6, 7, 8\u003c/p\u003e\n\u003cp\u003eOrdered:\u003c/p\u003e\n\u003cp\u003e6, 6, 7, 7, 7, 8, 8, 10\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eMedian\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\text{Median}=\\frac{7+7}{2}=7\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eMedian\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cstrong\u003eExample 3\u003c/strong\u003e\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: right\"\u003eScore \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_i)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003eFrequency \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(f_i)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3117em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ei\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e50\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e60\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e70\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e80\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e8\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e90\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e2\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e100\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e1\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eTotal data = 25, so the median position is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{25+1}{2}=13\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe 13th datum is \u003cstrong\u003e70\u003c/strong\u003e, so the median is \u003cstrong\u003e70\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"c-measures-of-data-spread\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-measures-of-data-spread\" class=\"header-mark\" aria-label=\"Direct link to section: C. Measures of Data Spread\"\u003e\u003c/a\u003eC. Measures of Data Spread\u003c/h2\u003e\u003ch3 id=\"1-range\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-range\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Range\"\u003e\u003c/a\u003e1. Range\u003c/h3\u003e\u003cp\u003eThe range is the difference between the largest and smallest data values.\u003c/p\u003e\n\u003cp\u003eData:\u003c/p\u003e\n\u003cp\u003e150, 155, 160, 157, 158, 160, 155, 150\u003c/p\u003e\n\u003cp\u003eOrdered:\u003c/p\u003e\n\u003cp\u003e150, 150, 155, 155, 157, 158, 160, 160\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e160\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e150\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n160-150=10\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e160\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e150\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-quartiles\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-quartiles\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Quartiles\"\u003e\u003c/a\u003e2. Quartiles\u003c/h3\u003e\u003cp\u003eQuartiles divide ordered data into four equal parts:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003elower quartile \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003emiddle quartile \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003eupper quartile \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ_3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eSteps:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003esort the data,\u003c/li\u003e\n\u003cli\u003edetermine \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or median,\u003c/li\u003e\n\u003cli\u003edetermine \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ_1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e from the lower half,\u003c/li\u003e\n\u003cli\u003edetermine \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ_3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e from the upper half.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003eExample\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ea. 20, 35, 50, 45, 30, 30, 25, 40, 45, 30, 35\u003cbr\u003e\nb. 11, 13, 10, 10, 12, 15, 14, 12\u003c/p\u003e\n\u003cp\u003eOrdered:\u003c/p\u003e\n\u003cp\u003ea. 20, 25, 30, 30, 30, 35, 35, 40, 45, 45, 50\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e35\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e45\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nQ_1=30,\\quad Q_2=35,\\quad Q_3=45\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e35\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e45\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eb. 10, 10, 11, 12, 12, 13, 14, 15\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" 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class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13.5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2112em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat concludes this statistics material.\u003c/p\u003e\n", "language": "en" }, { "title" : "Pyramid", "date_published" : "2021-02-08T08:02:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/02/limas/", "url" : "https://www.aantriono.com/2021/02/limas/", "summary": " Introduction\nBasic Competencies Distinguish and determine the surface area and volume of solid figures such as cubes, cuboids, prisms, and pyramids. 4.9 Solve problems related to the surface area and volume of solid figures and their combinations. Learning ObjectivesAfter studying pyramid material, students are expected to be able to:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji6z6P-dXpPPe9CI-gCC7LebSWJDIGmbSSDb4BVduKAngJaVWkaLXWJt44ENzhjM_pQvYCBH5V6y-8TY1gAOU0sUlTHW3JOmmezWYdy7eAlNpaU0RUms332Yp4prFynK4cV8YqOJIvuHo/w640-h292/limas.png' alt=\"House-shaped pyramid\" \u003e\u003c/p\u003e\n\u003ch3 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h3\u003e\u003col start=\"9\"\u003e\n\u003cli\u003eDistinguish and determine the surface area and volume of solid figures such as cubes, cuboids, prisms, and pyramids.\n4.9 Solve problems related to the surface area and volume of solid figures and their combinations.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h3\u003e\u003cp\u003eAfter studying pyramid material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eRecognize and mention faces, edges, face diagonals, space diagonals, and the height of a pyramid.\u003c/li\u003e\n\u003cli\u003eDescribe pyramid nets.\u003c/li\u003e\n\u003cli\u003eDetermine the surface area of a pyramid.\u003c/li\u003e\n\u003cli\u003eFind the volume formula and calculate the volume of a pyramid.\u003c/li\u003e\n\u003cli\u003eCalculate how the volume changes when the edge size changes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h3\u003e\u003ciframe height=\"480\" src=\"https://drive.google.com/file/d/1MlmaQ6kTg90oAMwIzB9QJgZlB1XZwWBC/preview\" width=\"640\"\u003e\u003c/iframe\u003e\n\u003ch3 id=\"geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: GeoGebra Simulation\"\u003e\u003c/a\u003eGeoGebra Simulation\u003c/h3\u003e\u003cp\u003eTo better understand pyramid material, please use the GeoGebra simulation below. Move the base and height sliders to the right or left.\u003c/p\u003e\n\u003ciframe height=\"433px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/pytqyqfg/width/650/height/433/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Surface area and volume of a pyramid\" width=\"650px\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h3\u003e\u003cp\u003eThe evaluation section is available in the full learning material that accompanies this article.\u003c/p\u003e\n", "language": "en" }, { "title" : "Prism", "date_published" : "2021-02-01T16:46:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/prisma/", "url" : "https://www.aantriono.com/en/prisma/", "summary": " IntroductionThis lesson covers the definition of a prism, its parts, diagonals, nets, surface area, volume, and volume changes.\nBasic Competencies3.9 Distinguish and determine the surface area and volume of polyhedra such as cubes, cuboids, prisms, and pyramids.\n4.9 Solve problems related to the surface area and volume of those solids and their combinations.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThis lesson covers the definition of a prism, its parts, diagonals, nets, surface area, volume, and volume changes.\u003c/p\u003e\n\u003ch2 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h2\u003e\u003cp\u003e3.9 Distinguish and determine the surface area and volume of polyhedra such as cubes, cuboids, prisms, and pyramids.\u003c/p\u003e\n\u003cp\u003e4.9 Solve problems related to the surface area and volume of those solids and their combinations.\u003c/p\u003e\n\u003ch2 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying this material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eIdentify faces, edges, plane diagonals, space diagonals, diagonal planes, and prism height.\u003c/li\u003e\n\u003cli\u003eDraw prism nets.\u003c/li\u003e\n\u003cli\u003eDetermine the surface area of a prism.\u003c/li\u003e\n\u003cli\u003eDerive and use the formula for the volume of a prism.\u003c/li\u003e\n\u003cli\u003eDetermine how prism volume changes when the base or height changes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h2\u003e\u003ch3 id=\"1-definition-of-a-prism\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-definition-of-a-prism\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Definition of a Prism\"\u003e\u003c/a\u003e1. Definition of a Prism\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgw9Yi4pHgT00w5L-pv4mDszYO34upnuZrzvS-Q1SqTmr1nYzEPGjzlpVODGzCnzVQWY7DYYeK3Koi9HmImKzAdG_PwyvFLuTtjkeQ1-rWtyxc_8jzDaPuXlQIhOBKeExy9dlLpaMmpPf8/w320-h165/prisma1.png' alt=\"Right prism\" \u003e\u003c/p\u003e\n\u003cp\u003eA prism is a polyhedron with two congruent and parallel faces, and lateral edges that are parallel to one another.\u003c/p\u003e\n\u003cp\u003eBased on the direction of the lateral edges, prisms are divided into:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eRight prisms\u003c/strong\u003e, whose lateral edges are perpendicular to the base and top faces.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eOblique prisms\u003c/strong\u003e, whose lateral edges are not perpendicular to the base and top faces.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRxYrDmYSB2O6Z9YVC_XAOlqDifN_MDIUjK3VBzD1c2lUShpi-tC0GFt7Cf9Amu-IQatJL2N3dVn1Ikn3GpJ8XcCRVHe03ikQS9_tkAZEkha8UhpR4Da757EG-HadsSyROxD2qAIogFP0/w320-h201/prisma2.png' alt=\"Oblique prism\" \u003e\u003c/p\u003e\n\u003cp\u003ePrisms are named after the shape of their base, such as triangular prisms, rectangular prisms, pentagonal prisms, and so on.\u003c/p\u003e\n\u003ch3 id=\"2-parts-of-a-prism\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-parts-of-a-prism\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Parts of a Prism\"\u003e\u003c/a\u003e2. Parts of a Prism\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK0-0FjV6ip94tuExqZcNcp3g9i1eF3RmAz9TxLvtuwQfDbiuSePZG6tY7qMC9ILrnrY8cDsfhORohMugYqiVZQACq1IUJvdJYId8GPuFL2WTYTTXqyvou2XXH2eG-0KMtEE-Ms51IYl0/w266-h320/prisma3.png' alt=\"Parts of a prism\" \u003e\u003c/p\u003e\n\u003cp\u003eFor the rectangular prism \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABCD.EFGH\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord\"\u003e.\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eEFG\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ePoints \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA, B, C, D, E, F, G,\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eH\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the vertices.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABCD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the base face.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eEFGH\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eEFG\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the top face.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABFE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eBFE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBCGF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eBCGF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eCDHG\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eDH\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eADHE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eDH\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the lateral faces.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{BC}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{CD}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AD}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the base edges.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{EF}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{FG}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eFG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{GH}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{EH}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the top edges.\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AE}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{BF}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eBF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{CG}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eCG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{DH}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eDH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are the lateral edges.\u003c/li\u003e\n\u003cli\u003eThe prism height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(t)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the distance between the base and top faces. In a right prism, it equals the length of a lateral edge.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"3-plane-diagonals-space-diagonals-and-diagonal-planes\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-plane-diagonals-space-diagonals-and-diagonal-planes\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Plane Diagonals, Space Diagonals, and Diagonal Planes\"\u003e\u003c/a\u003e3. Plane Diagonals, Space Diagonals, and Diagonal Planes\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYdd_5FhpD04H8j_fYAQ9IclCEStCokTQw-tAULqdqJL1doVK-eVtYxOftAXqoyTayWVpRJSzrAJ78fLNyMkGlQxlbMkrBg-NvdzUOC0RIIhuw-WRA6d7SJy-b1fdKXeVCyoakbhSM-rI/w277-h320/prisma4.png' alt=\"Prism diagonals\" \u003e\u003c/p\u003e\n\u003cp\u003eIn prism \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eKLMNO.PQRST\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eMNO\u003c/span\u003e\u003cspan class=\"mord\"\u003e.\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003ePQRST\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{KM}, \\overline{LN}, \\overline{MO}, \\overline{LO},\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eMO\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{KN}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are diagonals of the base face.\u003c/li\u003e\n\u003cli\u003ePlane \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eP\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eKMRP\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eMRP\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a \u003cstrong\u003ediagonal plane\u003c/strong\u003e, a plane containing a diagonal of the base and a corresponding diagonal of the top face.\u003c/li\u003e\n\u003cli\u003eSegments \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eLT\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eT\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eLS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eKR\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are \u003cstrong\u003espace diagonals\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eImportant definitions:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eA \u003cstrong\u003eplane diagonal\u003c/strong\u003e joins two non-adjacent vertices on the same face.\u003c/li\u003e\n\u003cli\u003eA \u003cstrong\u003espace diagonal\u003c/strong\u003e joins two vertices that do not lie on the same lateral face.\u003c/li\u003e\n\u003cli\u003eA \u003cstrong\u003ediagonal plane\u003c/strong\u003e contains diagonals of the base and top faces.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"4-prism-nets\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-prism-nets\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Prism Nets\"\u003e\u003c/a\u003e4. Prism Nets\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGMlnN4OoxDXePx8_T75GFFVRRxMf3-mIO6zSWG9xT8wcLWsQYlZaP2Sb9RMrt_OzGXP-ueKX6xCNzLnJg2surENxEqRzna2knPH9c8q60BjwdrRcj4TWnguCbYEQSmp6hlLUbXlqvS4g/w320-h222/prisma10.png' alt=\"Prism net\" \u003e\u003c/p\u003e\n\u003cp\u003eTo make a prism net:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eBuild a prism model from cardboard.\u003c/li\u003e\n\u003cli\u003eCut along selected edges.\u003c/li\u003e\n\u003cli\u003eUnfold the model until all faces lie flat.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"5-surface-area-of-a-prism\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-surface-area-of-a-prism\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Surface Area of a Prism\"\u003e\u003c/a\u003e5. Surface Area of a Prism\u003c/h3\u003e\u003cp\u003eFor the triangular prism \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003cmi\u003eU\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ePQR.STU\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003ePQR\u003c/span\u003e\u003cspan class=\"mord\"\u003e.\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eST\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eU\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the surface area is the sum of two bases and all lateral faces:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003ebase area\u003c/mtext\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmtext\u003ebase perimeter\u003c/mtext\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003eprism height\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= 2 \\times \\text{base area} + \\text{base perimeter} \\times \\text{prism height}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.5em;vertical-align:-0.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1em;\"\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1em;\"\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase area\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase perimeter\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eprism height\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the formula is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003ebase area\u003c/mtext\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmtext\u003ebase perimeter\u003c/mtext\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003eheight\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL = 2 \\times \\text{base area} + \\text{base perimeter} \\times \\text{height}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase area\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase perimeter\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eheight\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"examples\" class=\"headerLink\"\u003e\n \u003ca href=\"#examples\" class=\"header-mark\" aria-label=\"Direct link to section: Examples\"\u003e\u003c/a\u003eExamples\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eA triangular prism has a right-triangle base with side lengths \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm. Its height is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm. Find the surface area.\u003c/li\u003e\n\u003cli\u003eFind the surface area of the following prisms.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQqvjaRWulc2DhewSVoPM3BZ3iEXotBZDv418g13TlKPA8EaP4RwNygkmApq1iiudsQNSllXAj23GA_SEtqkfHvvdyCN4zMiE0UknGE_Vh-VPRSf7EveHpFHv8SmX5mOpKVqxpDx9OpAk/w200-h175/prisma9.png' alt=\"Trapezoidal prism\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtkGUinCxPqJzqj3r8j1VAcp460cLruUZMhLxSnby6wVEy7Mi8hkLy9hkPDNfQzHcQ1fswuYfgMdpsZWaVijuOdwCREmC_TGx69IEjYhPsjl0WH2XzopjFmx5E4-nkKwbrq9cuq8xm5E8/w200-h159/prisma8.png' alt=\"Triangular prism\" \u003e\u003c/p\u003e\n\u003ch4 id=\"solutions\" class=\"headerLink\"\u003e\n \u003ca href=\"#solutions\" class=\"header-mark\" aria-label=\"Direct link to section: Solutions\"\u003e\u003c/a\u003eSolutions\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003ebase area\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmtext\u003ebase perimeter\u003c/mtext\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003eheight\u003c/mtext\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e108\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e216\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e324\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= (2 \\times \\text{base area}) + (\\text{base perimeter} \\times \\text{height}) \\\\\n \u0026amp;= 2 \\times \\left(\\frac{1}{2} \\times 12 \\times 9\\right) + (12 + 9 + 15) \\times 6 \\\\\n \u0026amp;= 108 + 216 \\\\\n \u0026amp;= 324\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.2em;vertical-align:-3.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.85em;\"\u003e\u003cspan style=\"top:-6.46em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.35em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.26em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.76em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.35em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.85em;\"\u003e\u003cspan style=\"top:-6.46em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase area\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase perimeter\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eheight\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.35em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.26em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e108\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e216\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.76em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e324\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.35em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the surface area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e324\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e324\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e324\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eTrapezoidal prism:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eBase area\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e78\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\text{Base area} = \\frac{1}{2} \\times (4 + 9) \\times 12 = 78\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eBase area\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e78\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e78\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e156\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e380\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e536\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= (2 \\times 78) + (4 + 13 + 9 + 12) \\times 10 \\\\\n \u0026amp;= 156 + 380 \\\\\n \u0026amp;= 536\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e78\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e156\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e380\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e536\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the surface area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e536\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e536\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e536\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eTriangular prism:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e72\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e84\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= 2 \\times \\left(\\frac{1}{2} \\times 3 \\times 4\\right) + (3 + 4 + 5) \\times 6 \\\\\n \u0026amp;= 12 + 72 \\\\\n \u0026amp;= 84\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.7em;vertical-align:-2.6em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1em;\"\u003e\u003cspan style=\"top:-5.1em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.51em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.6em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1em;\"\u003e\u003cspan style=\"top:-5.1em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.01em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e72\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.51em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.45em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e84\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.6em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the surface area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e84\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e84\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e84\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"6-volume-of-a-prism\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-volume-of-a-prism\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Volume of a Prism\"\u003e\u003c/a\u003e6. Volume of a Prism\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2843wJ4qrGj_v8e3nwRmFzj1itrRUBcgPpDbYDFltuy9GjGEIJBkFrFzAOP-qFr6qSTgw0-DGT5cJAh9UiUbJW0XAC1lr4vHl2dq2YOoiGQIIrdcXPeDNchGxrXiqKc_hQkfEDarfovw/w200-h199/prisma6.png' alt=\"Prism volume\" \u003e\u003c/p\u003e\n\u003cp\u003eThe volume formula is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmtext\u003ebase area\u003c/mtext\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003eheight\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV = \\text{base area} \\times \\text{height}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ebase area\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eheight\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"examples-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#examples-1\" class=\"header-mark\" aria-label=\"Direct link to section: Examples\"\u003e\u003c/a\u003eExamples\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eFind the volume of a prism with height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and a right-triangle base whose perpendicular sides are \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/li\u003e\n\u003cli\u003eFind the volume of a prism with base area \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e30\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/li\u003e\n\u003cli\u003eFind the volume of the prism shown below.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi84VvGz7CNGeLsVEJYWg4fshpH2N57snliFzDH_qBeyV6sS4IETPH8YWYjzrPv6AWAmbGb0IDGE2Q2Fb9f_pomDVrua8Fpv6qrs8nM2-xLR65MLfuhsA29jaXZlU_2G4-4vXACzK5Apuc/w320-h216/prisma7.png' alt=\"Pentagonal prism\" \u003e\u003c/p\u003e\n\u003ch4 id=\"solutions-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#solutions-1\" class=\"header-mark\" aria-label=\"Direct link to section: Solutions\"\u003e\u003c/a\u003eSolutions\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eBase area:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{1}{2} \\times 4 \\times 3 = 6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eVolume:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e36\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV = 6 \\times 6 = 36\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the volume is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e36\\text{ cm}^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV = 30 \\times 2 = 60\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the volume is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e60\\text{ cm}^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eConsider \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle ABL\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Since it is isosceles:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e36\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e64\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nLM^2 \u0026amp;= 10^2 - 6^2 \\\\\n \u0026amp;= 100 - 36 \\\\\n \u0026amp;= 64 \\\\\nLM \u0026amp;= 8\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0241em;vertical-align:-2.7621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e64\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eArea of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle ABL\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e48\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{1}{2} \\times 12 \\times 8 = 48\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e48\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince the base consists of 5 congruent triangles:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eV\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e48\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4,800\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nV \u0026amp;= 5 \\times 48 \\times 20 \\\\\n \u0026amp;= 4{,}800\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e48\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e800\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the volume is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4,800\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4{,}800\\text{ cm}^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e800\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"7-change-in-prism-volume\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-change-in-prism-volume\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Change in Prism Volume\"\u003e\u003c/a\u003e7. Change in Prism Volume\u003c/h3\u003e\u003cp\u003eThe volume of a prism depends on the area of its base and its height. If either changes, the volume changes as well.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003cp\u003eA triangular prism has a right-triangle base with side lengths \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and prism height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm. The base is enlarged so the corresponding sides become \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, while the height stays the same. Determine:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ethe ratio of the new volume to the original volume,\u003c/li\u003e\n\u003cli\u003ethe amount of volume change.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h4\u003e\u003cp\u003eVolume ratio:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsub\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" 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class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" 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style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.836em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.9838em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5801em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5801em;\"\u003e\u003cspan style=\"top:-2.2649em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.735em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0801em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.2363em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.5801em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4764em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV_2 : V_1 = 4 : 1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eChange in volume:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nV_2 - V_1 \u0026amp;= 4V_1 - V_1 \\\\\n \u0026amp;= 3V_1 \\\\\n \u0026amp;= 3 \\times \\frac{1}{2} \\times 3 \\times 4 \\times 10 \\\\\n \u0026amp;= 180\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.8074em;vertical-align:-3.1537em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6537em;\"\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6537em;\"\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the change in volume is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180\\text{ cm}^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch2 id=\"geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: GeoGebra Simulation\"\u003e\u003c/a\u003eGeoGebra Simulation\u003c/h2\u003e\u003cp\u003eUse the following simulation to explore prism surface area and volume. Move the sliders for the triangle base, triangle height, and prism height.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/hfnfsrwy/width/650/height/400/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Luas Permukaan dan Volume Prisma\" width=\"650px\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eA triangular prism has a right-triangle base with perpendicular sides \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e24\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm. If the prism height is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e24\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, find the surface area.\u003c/li\u003e\n\u003cli\u003eFind the volume of each prism in the following figures.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgireyl61UPxs4Vg91K_CNWbN0KIRtDMhhwv6BCiIZm-NrqPenjUmzAnm588tGkWwNByQ1rBNTKWI5DsyPm4fPpW3Q4JHSLrNn-sO50O-7VtcFgVK_Hi1Zyiurh9W7R8lneSfB5iPSpVg/w200-h199/prisma11.png' alt=\"Prism evaluation 1\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_3vdc2E_CQt1RmuUgO7059ptgEpZovRUhBsrEGeadr-40PhhKAikHZeVhwekBqXFrLBx8r40tiWD3qre_JAqeUdvT2tkpMqr2FaG8O8dchWuUiryL0cqTeqQaG4Dw4noCIB7V-dz1A7w/w170-h200/prisma12.png' alt=\"Prism evaluation 2\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0gblXbQjPIV6d9H1qx11G7CLFufYNJ-6npd3ubCzmieitZ5i5SZzNyt2XjVQn4aJGQRUUNw63efB8vG00sMMdeVU8xm66tFataQkiv7VinIEwm3aBKTx9UkM9mgI7OGV0iJAWInZA7vg/w200-h168/prisma13.png' alt=\"Prism evaluation 3\" \u003e\u003c/p\u003e\n\u003cp\u003eThat completes this prism lesson. Corrections and suggestions can be shared through the available feedback channel.\u003c/p\u003e\n", "language": "en" }, { "title" : "Similarity and Congruence", "date_published" : "2021-01-29T16:37:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kesebangunan-dan-kekongruenan/", "url" : "https://www.aantriono.com/en/kesebangunan-dan-kekongruenan/", "summary": " IntroductionThis lesson discusses two important geometry ideas: similarity and congruence. Both are used to compare figures based on shape, size, side relationships, and angle relationships.\nBasic Competencies Explain and determine similarity and congruence among plane figures. Solve problems related to similarity and congruence among plane figures. Learning ObjectivesAfter studying this material, students are expected to be able to:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThis lesson discusses two important geometry ideas: \u003cstrong\u003esimilarity\u003c/strong\u003e and \u003cstrong\u003econgruence\u003c/strong\u003e. Both are used to compare figures based on shape, size, side relationships, and angle relationships.\u003c/p\u003e\n\u003ch2 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eExplain and determine similarity and congruence among plane figures.\u003c/li\u003e\n\u003cli\u003eSolve problems related to similarity and congruence among plane figures.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying this material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003edistinguish similar and non-similar figures,\u003c/li\u003e\n\u003cli\u003edistinguish congruent and non-congruent figures,\u003c/li\u003e\n\u003cli\u003edetermine unknown side lengths in similar figures,\u003c/li\u003e\n\u003cli\u003estate the conditions for congruent triangles,\u003c/li\u003e\n\u003cli\u003eprove that two triangles are similar,\u003c/li\u003e\n\u003cli\u003eapply similarity ratios to solve geometry problems.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"a-similar-plane-figures\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-similar-plane-figures\" class=\"header-mark\" aria-label=\"Direct link to section: A. Similar Plane Figures\"\u003e\u003c/a\u003eA. Similar Plane Figures\u003c/h2\u003e\u003ch3 id=\"1-two-similar-plane-figures\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-two-similar-plane-figures\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Two Similar Plane Figures\"\u003e\u003c/a\u003e1. Two Similar Plane Figures\u003c/h3\u003e\u003cp\u003eConsider the following figure.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ZKZ8mIt3mtQ7IcEhkZABSwGgu2UkpldYZYeBiQcblMFXUNzkLYDWy9J5DEcYJvHfEtVufEPRwgPVCKQDBixAWxnS-iRk0Gt7vRX6RUMGLfcJg1Npb6zBINqZf3gJb5hXBNuIK1oQ4Ok/w320-h192/kesebangunan1.png\" title=\"Similar plane figures\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ZKZ8mIt3mtQ7IcEhkZABSwGgu2UkpldYZYeBiQcblMFXUNzkLYDWy9J5DEcYJvHfEtVufEPRwgPVCKQDBixAWxnS-iRk0Gt7vRX6RUMGLfcJg1Npb6zBINqZf3gJb5hXBNuIK1oQ4Ok/w320-h192/kesebangunan1.png\" data-sub-html=\"\u003ch2\u003eSimilar plane figures\u003c/h2\u003e\u003cp\u003eSimilar plane figures\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ZKZ8mIt3mtQ7IcEhkZABSwGgu2UkpldYZYeBiQcblMFXUNzkLYDWy9J5DEcYJvHfEtVufEPRwgPVCKQDBixAWxnS-iRk0Gt7vRX6RUMGLfcJg1Npb6zBINqZf3gJb5hXBNuIK1oQ4Ok/w320-h192/kesebangunan1.png' alt=\"Similar plane figures\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSimilar plane figures\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ZKZ8mIt3mtQ7IcEhkZABSwGgu2UkpldYZYeBiQcblMFXUNzkLYDWy9J5DEcYJvHfEtVufEPRwgPVCKQDBixAWxnS-iRk0Gt7vRX6RUMGLfcJg1Npb6zBINqZf3gJb5hXBNuIK1oQ4Ok/s645/kesebangunan1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ZKZ8mIt3mtQ7IcEhkZABSwGgu2UkpldYZYeBiQcblMFXUNzkLYDWy9J5DEcYJvHfEtVufEPRwgPVCKQDBixAWxnS-iRk0Gt7vRX6RUMGLfcJg1Npb6zBINqZf3gJb5hXBNuIK1oQ4Ok/s645/kesebangunan1.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eSuppose rectangles \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABCD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ePQRS\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003ePQRS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e have corresponding sides:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{PQ}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ePQ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1944em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{BC}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{QR}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1944em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation 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style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{RS}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eRS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AD}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{PS}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003ePS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eTheir side ratios 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2408em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eRS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003ePS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe corresponding angles are also equal:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle A=\\angle P,\\quad \\angle B=\\angle Q,\\quad \\angle C=\\angle R,\\quad \\angle D=\\angle S\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eR\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e≃\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nABCD \\simeq PQRS\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≃\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003ePQRS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTwo plane figures are \u003cstrong\u003esimilar\u003c/strong\u003e if:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ecorresponding sides are proportional,\u003c/li\u003e\n\u003cli\u003ecorresponding angles are equal.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eConsider the two kites below.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje57Ka8U_khSvjjx7wqeWJL_dJWzwjCIuDvLXqw82F_MUHDJ2fme2Cw9DMbOoLKHMqTZ2BlI4EidMd6-RdPJ8oi19zKMcbF8SjzOzKGbEYjlIDRi0CBtP25zMCEkTPdM31kekHGSGokyU/w320-h234/kesebangunan2.png\" title=\"Similar kites\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje57Ka8U_khSvjjx7wqeWJL_dJWzwjCIuDvLXqw82F_MUHDJ2fme2Cw9DMbOoLKHMqTZ2BlI4EidMd6-RdPJ8oi19zKMcbF8SjzOzKGbEYjlIDRi0CBtP25zMCEkTPdM31kekHGSGokyU/w320-h234/kesebangunan2.png\" data-sub-html=\"\u003ch2\u003eSimilar kites\u003c/h2\u003e\u003cp\u003eSimilar kites\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje57Ka8U_khSvjjx7wqeWJL_dJWzwjCIuDvLXqw82F_MUHDJ2fme2Cw9DMbOoLKHMqTZ2BlI4EidMd6-RdPJ8oi19zKMcbF8SjzOzKGbEYjlIDRi0CBtP25zMCEkTPdM31kekHGSGokyU/w320-h234/kesebangunan2.png' alt=\"Similar kites\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSimilar kites\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje57Ka8U_khSvjjx7wqeWJL_dJWzwjCIuDvLXqw82F_MUHDJ2fme2Cw9DMbOoLKHMqTZ2BlI4EidMd6-RdPJ8oi19zKMcbF8SjzOzKGbEYjlIDRi0CBtP25zMCEkTPdM31kekHGSGokyU/s424/kesebangunan2.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje57Ka8U_khSvjjx7wqeWJL_dJWzwjCIuDvLXqw82F_MUHDJ2fme2Cw9DMbOoLKHMqTZ2BlI4EidMd6-RdPJ8oi19zKMcbF8SjzOzKGbEYjlIDRi0CBtP25zMCEkTPdM31kekHGSGokyU/s424/kesebangunan2.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eUsing the properties of kites, the corresponding angles are equal and the corresponding sides have the same ratio:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{BC}{FG}=\\frac{DC}{GH}=\\frac{AD}{EH}=\\frac{AB}{EF}=\\frac{3}{2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eFG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e≃\u003c/mo\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nABCD \\simeq EFGH\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≃\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eEFG\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-two-figures-with-the-same-shape-and-size\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-two-figures-with-the-same-shape-and-size\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Two Figures with the Same Shape and Size\"\u003e\u003c/a\u003e2. Two Figures with the Same Shape and Size\u003c/h3\u003e\u003cp\u003eConsider two identical banknotes.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI1ir1U_jQodqelyRcsIuJMQvycYjp_TN6QGbudoAvnoztN3NyQjYDOqVNMUuzHhglobGeqC8oaVO54v1HyjxelMFc26hK8h7px-WRxJl7sOa_KTCyPY1QtFmSZWA_2YAENi8ookan9aU/s320/kekongruenan.jpg\" title=\"Example of congruent figures\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI1ir1U_jQodqelyRcsIuJMQvycYjp_TN6QGbudoAvnoztN3NyQjYDOqVNMUuzHhglobGeqC8oaVO54v1HyjxelMFc26hK8h7px-WRxJl7sOa_KTCyPY1QtFmSZWA_2YAENi8ookan9aU/s320/kekongruenan.jpg\" data-sub-html=\"\u003ch2\u003eExample of congruent figures\u003c/h2\u003e\u003cp\u003eExample of congruent figures\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI1ir1U_jQodqelyRcsIuJMQvycYjp_TN6QGbudoAvnoztN3NyQjYDOqVNMUuzHhglobGeqC8oaVO54v1HyjxelMFc26hK8h7px-WRxJl7sOa_KTCyPY1QtFmSZWA_2YAENi8ookan9aU/s320/kekongruenan.jpg' alt=\"Example of congruent figures\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eExample of congruent figures\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI1ir1U_jQodqelyRcsIuJMQvycYjp_TN6QGbudoAvnoztN3NyQjYDOqVNMUuzHhglobGeqC8oaVO54v1HyjxelMFc26hK8h7px-WRxJl7sOa_KTCyPY1QtFmSZWA_2YAENi8ookan9aU/s598/kekongruenan.jpg\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI1ir1U_jQodqelyRcsIuJMQvycYjp_TN6QGbudoAvnoztN3NyQjYDOqVNMUuzHhglobGeqC8oaVO54v1HyjxelMFc26hK8h7px-WRxJl7sOa_KTCyPY1QtFmSZWA_2YAENi8ookan9aU/s598/kekongruenan.jpg\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf all corresponding sides are equal and all corresponding angles are equal, then the figures are \u003cstrong\u003econgruent\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eTwo figures are congruent if:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ecorresponding sides are equal in length,\u003c/li\u003e\n\u003cli\u003ecorresponding angles are equal.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLPU8t53gBSAUo-wNN_dRqS171cf9PLgMBNC6f-XhxAXfMVkScqOfza-aUt3Co5BqMkgJbZ_D0Y_jHk51cdxs0KS_r09fK4ZhIy1Le6UtZ_L9RxHiYKlh6CpIIF6WeYwnOp0ro5uFSPME/s320/kongruen.png\" title=\"Congruent rhombuses\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLPU8t53gBSAUo-wNN_dRqS171cf9PLgMBNC6f-XhxAXfMVkScqOfza-aUt3Co5BqMkgJbZ_D0Y_jHk51cdxs0KS_r09fK4ZhIy1Le6UtZ_L9RxHiYKlh6CpIIF6WeYwnOp0ro5uFSPME/s320/kongruen.png\" data-sub-html=\"\u003ch2\u003eCongruent rhombuses\u003c/h2\u003e\u003cp\u003eCongruent rhombuses\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLPU8t53gBSAUo-wNN_dRqS171cf9PLgMBNC6f-XhxAXfMVkScqOfza-aUt3Co5BqMkgJbZ_D0Y_jHk51cdxs0KS_r09fK4ZhIy1Le6UtZ_L9RxHiYKlh6CpIIF6WeYwnOp0ro5uFSPME/s320/kongruen.png' alt=\"Congruent rhombuses\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCongruent rhombuses\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLPU8t53gBSAUo-wNN_dRqS171cf9PLgMBNC6f-XhxAXfMVkScqOfza-aUt3Co5BqMkgJbZ_D0Y_jHk51cdxs0KS_r09fK4ZhIy1Le6UtZ_L9RxHiYKlh6CpIIF6WeYwnOp0ro5uFSPME/s515/kongruen.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLPU8t53gBSAUo-wNN_dRqS171cf9PLgMBNC6f-XhxAXfMVkScqOfza-aUt3Co5BqMkgJbZ_D0Y_jHk51cdxs0KS_r09fK4ZhIy1Le6UtZ_L9RxHiYKlh6CpIIF6WeYwnOp0ro5uFSPME/s515/kongruen.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf all corresponding sides of the two rhombuses have the same length and all corresponding angles are equal, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nABCD \\cong EFGH\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eEFG\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-finding-an-unknown-side\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-finding-an-unknown-side\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Finding an Unknown Side\"\u003e\u003c/a\u003e3. Finding an Unknown Side\u003c/h3\u003e\u003cp\u003eSimilarity can be used to find an unknown side length.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJQBiMpX0atYxR0E3PN_I2QPKK-cdNB-Mbbjro5XNseyXJom6WbsdEQXQjyNo7oxhkSaR_PP-7gYFCXzoMpp1bG3WguXom50dilJKFoF1MgNfM6bMWu134jO-eqjbER6mjlLdoRJv3TmQ/s320/kesebangunan3.png\" title=\"Finding unknown sides in similar figures\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJQBiMpX0atYxR0E3PN_I2QPKK-cdNB-Mbbjro5XNseyXJom6WbsdEQXQjyNo7oxhkSaR_PP-7gYFCXzoMpp1bG3WguXom50dilJKFoF1MgNfM6bMWu134jO-eqjbER6mjlLdoRJv3TmQ/s320/kesebangunan3.png\" data-sub-html=\"\u003ch2\u003eFinding unknown sides in similar figures\u003c/h2\u003e\u003cp\u003eFinding unknown sides in similar figures\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJQBiMpX0atYxR0E3PN_I2QPKK-cdNB-Mbbjro5XNseyXJom6WbsdEQXQjyNo7oxhkSaR_PP-7gYFCXzoMpp1bG3WguXom50dilJKFoF1MgNfM6bMWu134jO-eqjbER6mjlLdoRJv3TmQ/s320/kesebangunan3.png' alt=\"Finding unknown sides in similar figures\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eFinding unknown sides in similar figures\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJQBiMpX0atYxR0E3PN_I2QPKK-cdNB-Mbbjro5XNseyXJom6WbsdEQXQjyNo7oxhkSaR_PP-7gYFCXzoMpp1bG3WguXom50dilJKFoF1MgNfM6bMWu134jO-eqjbER6mjlLdoRJv3TmQ/s474/kesebangunan3.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJQBiMpX0atYxR0E3PN_I2QPKK-cdNB-Mbbjro5XNseyXJom6WbsdEQXQjyNo7oxhkSaR_PP-7gYFCXzoMpp1bG3WguXom50dilJKFoF1MgNfM6bMWu134jO-eqjbER6mjlLdoRJv3TmQ/s474/kesebangunan3.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eGiven:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{21}{9}=\\frac{7}{3}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{7}{x}=\\frac{7}{3}\\Rightarrow x=3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eand\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{14}{y}=\\frac{7}{3}\\Rightarrow y=6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2019em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=3\\text{ cm},\\quad y=6\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"b-similar-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-similar-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: B. Similar Triangles\"\u003e\u003c/a\u003eB. Similar Triangles\u003c/h2\u003e\u003ch3 id=\"1-conditions-for-similar-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-conditions-for-similar-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Conditions for Similar Triangles\"\u003e\u003c/a\u003e1. Conditions for Similar Triangles\u003c/h3\u003e\u003cp\u003eConsider the figure below.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjirwFZW0aUIfzMvtQs21NnXaXnrwM7UPEKhc8dyAVA0M7fgEozm6B8aX3Urjvd3Q5uT8jZm1ONywf9F2uQQ3jaT9itklYEWsE-Zg9lTBPXYw6xi6_yZ5gb6POedOlynCeeVK4BCtgXnWA/s320/kesebangunan4.png\" title=\"Similar triangles\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjirwFZW0aUIfzMvtQs21NnXaXnrwM7UPEKhc8dyAVA0M7fgEozm6B8aX3Urjvd3Q5uT8jZm1ONywf9F2uQQ3jaT9itklYEWsE-Zg9lTBPXYw6xi6_yZ5gb6POedOlynCeeVK4BCtgXnWA/s320/kesebangunan4.png\" data-sub-html=\"\u003ch2\u003eSimilar triangles\u003c/h2\u003e\u003cp\u003eSimilar triangles\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjirwFZW0aUIfzMvtQs21NnXaXnrwM7UPEKhc8dyAVA0M7fgEozm6B8aX3Urjvd3Q5uT8jZm1ONywf9F2uQQ3jaT9itklYEWsE-Zg9lTBPXYw6xi6_yZ5gb6POedOlynCeeVK4BCtgXnWA/s320/kesebangunan4.png' alt=\"Similar triangles\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSimilar triangles\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjirwFZW0aUIfzMvtQs21NnXaXnrwM7UPEKhc8dyAVA0M7fgEozm6B8aX3Urjvd3Q5uT8jZm1ONywf9F2uQQ3jaT9itklYEWsE-Zg9lTBPXYw6xi6_yZ5gb6POedOlynCeeVK4BCtgXnWA/s856/kesebangunan4.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjirwFZW0aUIfzMvtQs21NnXaXnrwM7UPEKhc8dyAVA0M7fgEozm6B8aX3Urjvd3Q5uT8jZm1ONywf9F2uQQ3jaT9itklYEWsE-Zg9lTBPXYw6xi6_yZ5gb6POedOlynCeeVK4BCtgXnWA/s856/kesebangunan4.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{AB}{DE}=\\frac{BC}{EF}=\\frac{AC}{DF}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eand the corresponding angles are equal, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≃\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle ABC \\simeq \\triangle DEF\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≃\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIn general, two triangles are similar if:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ecorresponding sides are proportional, or\u003c/li\u003e\n\u003cli\u003etwo pairs of corresponding angles are equal.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBQwbXpGi1ce029mlo_k2LmLej3MHyZiCU-Onq_Nc6Q7tAgfhB5gX4ukUVr3hVbr326TnbjQsupqCwcf_-DizYJ3TiJM_rfjnfyQ5lHsuLkLXZz7pvUslXwImQxFC5ERzE88wNpe-tzWI/w320-h208/kongruen1.png\" title=\"Side comparison in similar triangles\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBQwbXpGi1ce029mlo_k2LmLej3MHyZiCU-Onq_Nc6Q7tAgfhB5gX4ukUVr3hVbr326TnbjQsupqCwcf_-DizYJ3TiJM_rfjnfyQ5lHsuLkLXZz7pvUslXwImQxFC5ERzE88wNpe-tzWI/w320-h208/kongruen1.png\" data-sub-html=\"\u003ch2\u003eSide comparison in similar triangles\u003c/h2\u003e\u003cp\u003eSide comparison in similar triangles\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBQwbXpGi1ce029mlo_k2LmLej3MHyZiCU-Onq_Nc6Q7tAgfhB5gX4ukUVr3hVbr326TnbjQsupqCwcf_-DizYJ3TiJM_rfjnfyQ5lHsuLkLXZz7pvUslXwImQxFC5ERzE88wNpe-tzWI/w320-h208/kongruen1.png' alt=\"Side comparison in similar triangles\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSide comparison in similar triangles\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBQwbXpGi1ce029mlo_k2LmLej3MHyZiCU-Onq_Nc6Q7tAgfhB5gX4ukUVr3hVbr326TnbjQsupqCwcf_-DizYJ3TiJM_rfjnfyQ5lHsuLkLXZz7pvUslXwImQxFC5ERzE88wNpe-tzWI/s446/kongruen1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBQwbXpGi1ce029mlo_k2LmLej3MHyZiCU-Onq_Nc6Q7tAgfhB5gX4ukUVr3hVbr326TnbjQsupqCwcf_-DizYJ3TiJM_rfjnfyQ5lHsuLkLXZz7pvUslXwImQxFC5ERzE88wNpe-tzWI/s446/kongruen1.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{KL}{PQ}=\\frac{LM}{QR}=\\frac{KM}{PR}=\\frac{2}{3}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2408em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ePQ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2408em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003ePR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e≃\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle KLM \\simeq \\triangle PQR\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≃\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003ePQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-special-similarity-in-right-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-special-similarity-in-right-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Special Similarity in Right Triangles\"\u003e\u003c/a\u003e2. Special Similarity in Right Triangles\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRWDMxs_XheXiZ3UMOFiFfI2efQ0TT_u1ay_ji8aveskfxFN3V4-fNRt1IclkcdxORpzYksOv4U95c3Rtd0F4uDVQrD9u_4gIPbIC1uOv3rSAe-uBX0SexWH8M1-tYrKejpOU84bdGfuo/w200-h194/kongruen2.png\" title=\"Similarity in right triangles\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRWDMxs_XheXiZ3UMOFiFfI2efQ0TT_u1ay_ji8aveskfxFN3V4-fNRt1IclkcdxORpzYksOv4U95c3Rtd0F4uDVQrD9u_4gIPbIC1uOv3rSAe-uBX0SexWH8M1-tYrKejpOU84bdGfuo/w200-h194/kongruen2.png\" data-sub-html=\"\u003ch2\u003eSimilarity in right triangles\u003c/h2\u003e\u003cp\u003eSimilarity in right triangles\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRWDMxs_XheXiZ3UMOFiFfI2efQ0TT_u1ay_ji8aveskfxFN3V4-fNRt1IclkcdxORpzYksOv4U95c3Rtd0F4uDVQrD9u_4gIPbIC1uOv3rSAe-uBX0SexWH8M1-tYrKejpOU84bdGfuo/w200-h194/kongruen2.png' alt=\"Similarity in right triangles\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSimilarity in right triangles\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRWDMxs_XheXiZ3UMOFiFfI2efQ0TT_u1ay_ji8aveskfxFN3V4-fNRt1IclkcdxORpzYksOv4U95c3Rtd0F4uDVQrD9u_4gIPbIC1uOv3rSAe-uBX0SexWH8M1-tYrKejpOU84bdGfuo/s295/kongruen2.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRWDMxs_XheXiZ3UMOFiFfI2efQ0TT_u1ay_ji8aveskfxFN3V4-fNRt1IclkcdxORpzYksOv4U95c3Rtd0F4uDVQrD9u_4gIPbIC1uOv3rSAe-uBX0SexWH8M1-tYrKejpOU84bdGfuo/s295/kongruen2.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eFor right triangles, these useful relationships hold:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAD^2=BD\\times CD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB^2=BD\\times BC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAC^2=CD\\times CB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eCB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAcL4-UHAsf-KJuj9R3FAcyd7wTnIPp2DXRKCvNyy57r1XbGXTj127CwL9FzC0hSOVTjnUzqnuKTE6dDjJiIG4JHVIwId4jTPrhfkcPHhSbo8O_s74wJV9zN27j5rXcSJDE80-ZVRip1U/w200-h196/kongruen3.png\" title=\"Example of similar right triangles\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAcL4-UHAsf-KJuj9R3FAcyd7wTnIPp2DXRKCvNyy57r1XbGXTj127CwL9FzC0hSOVTjnUzqnuKTE6dDjJiIG4JHVIwId4jTPrhfkcPHhSbo8O_s74wJV9zN27j5rXcSJDE80-ZVRip1U/w200-h196/kongruen3.png\" data-sub-html=\"\u003ch2\u003eExample of similar right triangles\u003c/h2\u003e\u003cp\u003eExample of similar right triangles\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAcL4-UHAsf-KJuj9R3FAcyd7wTnIPp2DXRKCvNyy57r1XbGXTj127CwL9FzC0hSOVTjnUzqnuKTE6dDjJiIG4JHVIwId4jTPrhfkcPHhSbo8O_s74wJV9zN27j5rXcSJDE80-ZVRip1U/w200-h196/kongruen3.png' alt=\"Example of similar right triangles\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eExample of similar right triangles\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAcL4-UHAsf-KJuj9R3FAcyd7wTnIPp2DXRKCvNyy57r1XbGXTj127CwL9FzC0hSOVTjnUzqnuKTE6dDjJiIG4JHVIwId4jTPrhfkcPHhSbo8O_s74wJV9zN27j5rXcSJDE80-ZVRip1U/s287/kongruen3.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAcL4-UHAsf-KJuj9R3FAcyd7wTnIPp2DXRKCvNyy57r1XbGXTj127CwL9FzC0hSOVTjnUzqnuKTE6dDjJiIG4JHVIwId4jTPrhfkcPHhSbo8O_s74wJV9zN27j5rXcSJDE80-ZVRip1U/s287/kongruen3.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB=6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBC=8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nAC=\\sqrt{6^2+8^2}=10\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan 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style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.4\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" 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style=\"height:0.7667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e23.04\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4.8\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nBD=4.8\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4.8\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-finding-unknown-sides-in-similar-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-finding-unknown-sides-in-similar-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Finding Unknown Sides in Similar Triangles\"\u003e\u003c/a\u003e3. Finding Unknown Sides in Similar Triangles\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq72y5ddBnTZBVnwYLjgT9lZoyh39L88mA-fbOkqgd3lgGLG_pzgHf91fukGAu-5u4-xkgclQMUqKmQBH0Nx20aR_xWjddD-Q8Se4X-ZFXaAlk4iChFrjB2l8WOfsSrtqI4xFdUHRVd5g/w320-h170/kongruen4.png\" title=\"Finding side lengths in similar triangles\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq72y5ddBnTZBVnwYLjgT9lZoyh39L88mA-fbOkqgd3lgGLG_pzgHf91fukGAu-5u4-xkgclQMUqKmQBH0Nx20aR_xWjddD-Q8Se4X-ZFXaAlk4iChFrjB2l8WOfsSrtqI4xFdUHRVd5g/w320-h170/kongruen4.png\" data-sub-html=\"\u003ch2\u003eFinding side lengths in similar triangles\u003c/h2\u003e\u003cp\u003eFinding side lengths in similar triangles\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq72y5ddBnTZBVnwYLjgT9lZoyh39L88mA-fbOkqgd3lgGLG_pzgHf91fukGAu-5u4-xkgclQMUqKmQBH0Nx20aR_xWjddD-Q8Se4X-ZFXaAlk4iChFrjB2l8WOfsSrtqI4xFdUHRVd5g/w320-h170/kongruen4.png' alt=\"Finding side lengths in similar triangles\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eFinding side lengths in similar triangles\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq72y5ddBnTZBVnwYLjgT9lZoyh39L88mA-fbOkqgd3lgGLG_pzgHf91fukGAu-5u4-xkgclQMUqKmQBH0Nx20aR_xWjddD-Q8Se4X-ZFXaAlk4iChFrjB2l8WOfsSrtqI4xFdUHRVd5g/s474/kongruen4.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq72y5ddBnTZBVnwYLjgT9lZoyh39L88mA-fbOkqgd3lgGLG_pzgHf91fukGAu-5u4-xkgclQMUqKmQBH0Nx20aR_xWjddD-Q8Se4X-ZFXaAlk4iChFrjB2l8WOfsSrtqI4xFdUHRVd5g/s474/kongruen4.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≃\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle ABC \\simeq \\triangle DEF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≃\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{AB}{DE}=\\frac{BC}{EF}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7.5\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{12}{6}=\\frac{15}{EF}\n\\Rightarrow EF=7.5\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7.5\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAnother example:\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihUjZzch7oIfocLwS9nH6uvd4MGZkSOZdPPFATO2JHcNe87nAcBMwfiXLnhgFUisa7lrYe_YIp3qTD60BUSBA0AKtz_IH6hJTbbD43ni5epjU7ZtAIBJDI768eRYumI_LShxhDImGTe4c/w198-h200/kongruen5.png\" title=\"Similar triangles formed by parallel lines\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihUjZzch7oIfocLwS9nH6uvd4MGZkSOZdPPFATO2JHcNe87nAcBMwfiXLnhgFUisa7lrYe_YIp3qTD60BUSBA0AKtz_IH6hJTbbD43ni5epjU7ZtAIBJDI768eRYumI_LShxhDImGTe4c/w198-h200/kongruen5.png\" data-sub-html=\"\u003ch2\u003eSimilar triangles formed by parallel lines\u003c/h2\u003e\u003cp\u003eSimilar triangles formed by parallel lines\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihUjZzch7oIfocLwS9nH6uvd4MGZkSOZdPPFATO2JHcNe87nAcBMwfiXLnhgFUisa7lrYe_YIp3qTD60BUSBA0AKtz_IH6hJTbbD43ni5epjU7ZtAIBJDI768eRYumI_LShxhDImGTe4c/w198-h200/kongruen5.png' alt=\"Similar triangles formed by parallel lines\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSimilar triangles formed by parallel lines\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihUjZzch7oIfocLwS9nH6uvd4MGZkSOZdPPFATO2JHcNe87nAcBMwfiXLnhgFUisa7lrYe_YIp3qTD60BUSBA0AKtz_IH6hJTbbD43ni5epjU7ZtAIBJDI768eRYumI_LShxhDImGTe4c/s293/kongruen5.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihUjZzch7oIfocLwS9nH6uvd4MGZkSOZdPPFATO2JHcNe87nAcBMwfiXLnhgFUisa7lrYe_YIp3qTD60BUSBA0AKtz_IH6hJTbbD43ni5epjU7ZtAIBJDI768eRYumI_LShxhDImGTe4c/s293/kongruen5.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eBecause \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e≃\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle AED \\simeq \\triangle BCE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≃\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eBCE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{AD}{BC}=\\frac{AE}{CE}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eCE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6.75\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{9}{16}=\\frac{AE}{12}\n\\Rightarrow AE=6.75\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.75\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"4-parallel-lines-on-triangle-sides\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-parallel-lines-on-triangle-sides\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Parallel Lines on Triangle Sides\"\u003e\u003c/a\u003e4. Parallel Lines on Triangle Sides\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOwyY4A6iCEAKuubHHgifqXjVOdsmyqNAcFzrv_l_USj2QlSQ8pKebR_Bly0vVtCgT93W4auN2owWNcAbRDJWQfbC7c6o5EmEsrz2jeKbdvD1EzPcR_PmF3aXsJHRrIZPxlXHF19nO7LA/w320-h174/kongruen6.png\" title=\"Parallel lines on a triangle\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOwyY4A6iCEAKuubHHgifqXjVOdsmyqNAcFzrv_l_USj2QlSQ8pKebR_Bly0vVtCgT93W4auN2owWNcAbRDJWQfbC7c6o5EmEsrz2jeKbdvD1EzPcR_PmF3aXsJHRrIZPxlXHF19nO7LA/w320-h174/kongruen6.png\" data-sub-html=\"\u003ch2\u003eParallel lines on a triangle\u003c/h2\u003e\u003cp\u003eParallel lines on a triangle\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOwyY4A6iCEAKuubHHgifqXjVOdsmyqNAcFzrv_l_USj2QlSQ8pKebR_Bly0vVtCgT93W4auN2owWNcAbRDJWQfbC7c6o5EmEsrz2jeKbdvD1EzPcR_PmF3aXsJHRrIZPxlXHF19nO7LA/w320-h174/kongruen6.png' alt=\"Parallel lines on a triangle\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eParallel lines on a triangle\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOwyY4A6iCEAKuubHHgifqXjVOdsmyqNAcFzrv_l_USj2QlSQ8pKebR_Bly0vVtCgT93W4auN2owWNcAbRDJWQfbC7c6o5EmEsrz2jeKbdvD1EzPcR_PmF3aXsJHRrIZPxlXHF19nO7LA/s481/kongruen6.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOwyY4A6iCEAKuubHHgifqXjVOdsmyqNAcFzrv_l_USj2QlSQ8pKebR_Bly0vVtCgT93W4auN2owWNcAbRDJWQfbC7c6o5EmEsrz2jeKbdvD1EzPcR_PmF3aXsJHRrIZPxlXHF19nO7LA/s481/kongruen6.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo\u003e∥\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eDE\\parallel AB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≃\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle DEC \\simeq \\triangle ABC\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eEC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≃\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{CD}{AC}=\\frac{CE}{BC}=\\frac{DE}{AB}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eCE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThis often leads to:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{d}{a}=\\frac{c}{b}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0574em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-4\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9lsjToGeqTzQv4CQUmLPG5gG54wIU_RBF3s3vt8quu8Kg-gaQVKKOjEdfRlRK6-6kzgpVsvBhj-FRxrNyUacpmEAd_T6s0sBm9-bUR3ecndZL5MplGR2lcp_56Fz1Bv5n1fxrv8gdJmE/w320-h198/kongruen7.png\" title=\"Example of parallel lines in a triangle\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9lsjToGeqTzQv4CQUmLPG5gG54wIU_RBF3s3vt8quu8Kg-gaQVKKOjEdfRlRK6-6kzgpVsvBhj-FRxrNyUacpmEAd_T6s0sBm9-bUR3ecndZL5MplGR2lcp_56Fz1Bv5n1fxrv8gdJmE/w320-h198/kongruen7.png\" data-sub-html=\"\u003ch2\u003eExample of parallel lines in a triangle\u003c/h2\u003e\u003cp\u003eExample of parallel lines in a triangle\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9lsjToGeqTzQv4CQUmLPG5gG54wIU_RBF3s3vt8quu8Kg-gaQVKKOjEdfRlRK6-6kzgpVsvBhj-FRxrNyUacpmEAd_T6s0sBm9-bUR3ecndZL5MplGR2lcp_56Fz1Bv5n1fxrv8gdJmE/w320-h198/kongruen7.png' alt=\"Example of parallel lines in a triangle\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eExample of parallel lines in a triangle\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9lsjToGeqTzQv4CQUmLPG5gG54wIU_RBF3s3vt8quu8Kg-gaQVKKOjEdfRlRK6-6kzgpVsvBhj-FRxrNyUacpmEAd_T6s0sBm9-bUR3ecndZL5MplGR2lcp_56Fz1Bv5n1fxrv8gdJmE/s426/kongruen7.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9lsjToGeqTzQv4CQUmLPG5gG54wIU_RBF3s3vt8quu8Kg-gaQVKKOjEdfRlRK6-6kzgpVsvBhj-FRxrNyUacpmEAd_T6s0sBm9-bUR3ecndZL5MplGR2lcp_56Fz1Bv5n1fxrv8gdJmE/s426/kongruen7.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e∥\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{PT}\\parallel \\overline{QS}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1333em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003ePT\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eQS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1944em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{QS}{PT}=\\frac{QR}{PQ+QR}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003ePT\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eQS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2408em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ePQ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{3}{4}=\\frac{QR}{2+QR}\n\\Rightarrow QR=6\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2408em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.00773em;\"\u003eQR\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAnd:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{QS}{PT}=\\frac{RS}{ST+RS}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003ePT\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eQS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1297em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eST\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eRS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eRS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmrow\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003eS\u003c/mi\u003e\u003cmi\u003eT\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{3}{4}=\\frac{4}{ST+4}\n\\Rightarrow ST=\\frac{4}{3}\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eST\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eST\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"5-word-problems-involving-similarity\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-word-problems-involving-similarity\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Word Problems Involving Similarity\"\u003e\u003c/a\u003e5. Word Problems Involving Similarity\u003c/h3\u003e\u003cp\u003eTwo common applications are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003efinding the height of a pole using a smaller reference object,\u003c/li\u003e\n\u003cli\u003efinding missing dimensions in a photo and frame setup.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eFor the pole problem:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{2}{t}=\\frac{3}{9}\\Rightarrow t=6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the height of the pole is \u003cstrong\u003e6 m\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eFor the frame problem:\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbJAGPh8lQCrQsTcTdJrPk4FAmZJKcDQfBrIB3Hl4jhqAOx0c36AExis7sJQT1tlwZRwhAH5quh4gBPtXrX-pi02SRgUhAROFvaXqFIadRd-peX7ab6EuTWhUolY04D6Gx-re7lebILI/w302-h320/kongruen12.png\" title=\"Frame similarity problem\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbJAGPh8lQCrQsTcTdJrPk4FAmZJKcDQfBrIB3Hl4jhqAOx0c36AExis7sJQT1tlwZRwhAH5quh4gBPtXrX-pi02SRgUhAROFvaXqFIadRd-peX7ab6EuTWhUolY04D6Gx-re7lebILI/w302-h320/kongruen12.png\" data-sub-html=\"\u003ch2\u003eFrame similarity problem\u003c/h2\u003e\u003cp\u003eFrame similarity problem\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbJAGPh8lQCrQsTcTdJrPk4FAmZJKcDQfBrIB3Hl4jhqAOx0c36AExis7sJQT1tlwZRwhAH5quh4gBPtXrX-pi02SRgUhAROFvaXqFIadRd-peX7ab6EuTWhUolY04D6Gx-re7lebILI/w302-h320/kongruen12.png' alt=\"Frame similarity problem\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eFrame similarity problem\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbJAGPh8lQCrQsTcTdJrPk4FAmZJKcDQfBrIB3Hl4jhqAOx0c36AExis7sJQT1tlwZRwhAH5quh4gBPtXrX-pi02SRgUhAROFvaXqFIadRd-peX7ab6EuTWhUolY04D6Gx-re7lebILI/s307/kongruen12.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbJAGPh8lQCrQsTcTdJrPk4FAmZJKcDQfBrIB3Hl4jhqAOx0c36AExis7sJQT1tlwZRwhAH5quh4gBPtXrX-pi02SRgUhAROFvaXqFIadRd-peX7ab6EuTWhUolY04D6Gx-re7lebILI/s307/kongruen12.png\u003c/a\u003e)\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e54\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e28\u003c/mn\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{54-x}{60}=\\frac{28}{40}=\\frac{7}{10}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e54\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e28\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e54\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e42\u003c/mn\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n54-x=42\\Rightarrow x=12\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e54\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e42\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the lower frame width is \u003cstrong\u003e12 cm\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"c-congruent-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-congruent-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: C. Congruent Triangles\"\u003e\u003c/a\u003eC. Congruent Triangles\u003c/h2\u003e\u003ch3 id=\"1-properties-of-congruent-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-properties-of-congruent-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Properties of Congruent Triangles\"\u003e\u003c/a\u003e1. Properties of Congruent Triangles\u003c/h3\u003e\u003cp\u003eCongruent triangles have:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ethe same shape,\u003c/li\u003e\n\u003cli\u003ethe same size,\u003c/li\u003e\n\u003cli\u003eequal corresponding side lengths,\u003c/li\u003e\n\u003cli\u003eequal corresponding angles.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY1KhS3uwpgCLsHzjagznwWF58gG_Dw7QFSMsrsDfj_WGW7iZvWyWussyL6v8nBXfviKaVpWGfDvL7dgXnFZb1i_RkHxSsiHi41QXX02UEAOTIS2f0dDDri5olOCeL0Vs0RI75v7uvReo/w320-h169/kongruen8.png\" title=\"Properties of congruent triangles\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY1KhS3uwpgCLsHzjagznwWF58gG_Dw7QFSMsrsDfj_WGW7iZvWyWussyL6v8nBXfviKaVpWGfDvL7dgXnFZb1i_RkHxSsiHi41QXX02UEAOTIS2f0dDDri5olOCeL0Vs0RI75v7uvReo/w320-h169/kongruen8.png\" data-sub-html=\"\u003ch2\u003eProperties of congruent triangles\u003c/h2\u003e\u003cp\u003eProperties of congruent triangles\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY1KhS3uwpgCLsHzjagznwWF58gG_Dw7QFSMsrsDfj_WGW7iZvWyWussyL6v8nBXfviKaVpWGfDvL7dgXnFZb1i_RkHxSsiHi41QXX02UEAOTIS2f0dDDri5olOCeL0Vs0RI75v7uvReo/w320-h169/kongruen8.png' alt=\"Properties of congruent triangles\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eProperties of congruent triangles\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY1KhS3uwpgCLsHzjagznwWF58gG_Dw7QFSMsrsDfj_WGW7iZvWyWussyL6v8nBXfviKaVpWGfDvL7dgXnFZb1i_RkHxSsiHi41QXX02UEAOTIS2f0dDDri5olOCeL0Vs0RI75v7uvReo/s514/kongruen8.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY1KhS3uwpgCLsHzjagznwWF58gG_Dw7QFSMsrsDfj_WGW7iZvWyWussyL6v8nBXfviKaVpWGfDvL7dgXnFZb1i_RkHxSsiHi41QXX02UEAOTIS2f0dDDri5olOCeL0Vs0RI75v7uvReo/s514/kongruen8.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch3 id=\"2-conditions-for-congruent-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-conditions-for-congruent-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Conditions for Congruent Triangles\"\u003e\u003c/a\u003e2. Conditions for Congruent Triangles\u003c/h3\u003e\u003cp\u003eTwo triangles are congruent if they satisfy one of these conditions:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eSSS\u003c/strong\u003e: all three corresponding sides are equal,\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSAS\u003c/strong\u003e: two corresponding sides and the included angle are equal,\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eASA\u003c/strong\u003e: two corresponding angles and the included side are equal.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"a-sss\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-sss\" class=\"header-mark\" aria-label=\"Direct link to section: a. SSS\"\u003e\u003c/a\u003ea. SSS\u003c/h4\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxfQu3wweo2717qoYWnybjxlCqoAU1jieZP4bA9JsNpI07JEL7-4F1rexBMDVTsufA8KoX5k3uYSTd54YES8VN9eiCuzW740S6rwisPAe3WTOlMT963KqTUbO0xMwtzYb7WKvqOnlLVIs/w320-h142/kongruen9.png\" title=\"Congruence by SSS\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxfQu3wweo2717qoYWnybjxlCqoAU1jieZP4bA9JsNpI07JEL7-4F1rexBMDVTsufA8KoX5k3uYSTd54YES8VN9eiCuzW740S6rwisPAe3WTOlMT963KqTUbO0xMwtzYb7WKvqOnlLVIs/w320-h142/kongruen9.png\" data-sub-html=\"\u003ch2\u003eCongruence by SSS\u003c/h2\u003e\u003cp\u003eCongruence by SSS\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxfQu3wweo2717qoYWnybjxlCqoAU1jieZP4bA9JsNpI07JEL7-4F1rexBMDVTsufA8KoX5k3uYSTd54YES8VN9eiCuzW740S6rwisPAe3WTOlMT963KqTUbO0xMwtzYb7WKvqOnlLVIs/w320-h142/kongruen9.png' alt=\"Congruence by SSS\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCongruence by SSS\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxfQu3wweo2717qoYWnybjxlCqoAU1jieZP4bA9JsNpI07JEL7-4F1rexBMDVTsufA8KoX5k3uYSTd54YES8VN9eiCuzW740S6rwisPAe3WTOlMT963KqTUbO0xMwtzYb7WKvqOnlLVIs/s480/kongruen9.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxfQu3wweo2717qoYWnybjxlCqoAU1jieZP4bA9JsNpI07JEL7-4F1rexBMDVTsufA8KoX5k3uYSTd54YES8VN9eiCuzW740S6rwisPAe3WTOlMT963KqTUbO0xMwtzYb7WKvqOnlLVIs/s480/kongruen9.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nAB=DE,\\quad BC=EF,\\quad AC=DF\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord 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class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle ABC \\cong \\triangle DEF\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"b-sas\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-sas\" class=\"header-mark\" aria-label=\"Direct link to section: b. SAS\"\u003e\u003c/a\u003eb. SAS\u003c/h4\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMSyW8WFjUkzKMd7T86mJudESYC9RrC4B7-fpllf1jDyyYbA49r75KpTDm22Bs6gvrMSLkgqvfovWJlCSKCxi6of0rHYjHFEulXAWcwSVaskj8nV_V3ILhciFM8TS8bIZ8n2n_5RyjD00/w320-h143/kongruen11.png\" title=\"Congruence by SAS\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMSyW8WFjUkzKMd7T86mJudESYC9RrC4B7-fpllf1jDyyYbA49r75KpTDm22Bs6gvrMSLkgqvfovWJlCSKCxi6of0rHYjHFEulXAWcwSVaskj8nV_V3ILhciFM8TS8bIZ8n2n_5RyjD00/w320-h143/kongruen11.png\" data-sub-html=\"\u003ch2\u003eCongruence by SAS\u003c/h2\u003e\u003cp\u003eCongruence by SAS\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMSyW8WFjUkzKMd7T86mJudESYC9RrC4B7-fpllf1jDyyYbA49r75KpTDm22Bs6gvrMSLkgqvfovWJlCSKCxi6of0rHYjHFEulXAWcwSVaskj8nV_V3ILhciFM8TS8bIZ8n2n_5RyjD00/w320-h143/kongruen11.png' alt=\"Congruence by SAS\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCongruence by SAS\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMSyW8WFjUkzKMd7T86mJudESYC9RrC4B7-fpllf1jDyyYbA49r75KpTDm22Bs6gvrMSLkgqvfovWJlCSKCxi6of0rHYjHFEulXAWcwSVaskj8nV_V3ILhciFM8TS8bIZ8n2n_5RyjD00/s448/kongruen11.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMSyW8WFjUkzKMd7T86mJudESYC9RrC4B7-fpllf1jDyyYbA49r75KpTDm22Bs6gvrMSLkgqvfovWJlCSKCxi6of0rHYjHFEulXAWcwSVaskj8nV_V3ILhciFM8TS8bIZ8n2n_5RyjD00/s448/kongruen11.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nAB=DE,\\quad AC=DF,\\quad \\angle BAC=\\angle 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class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle ABC \\cong \\triangle DEF\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"c-asa\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-asa\" class=\"header-mark\" aria-label=\"Direct link to section: c. ASA\"\u003e\u003c/a\u003ec. ASA\u003c/h4\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdRI1oBtJiRc5ROsGFOdUXCbyxMRGQ89OnC2q8fDMU8aEXlpGaenlVPDSqW9uiRdPHnvyun-BJGZelDfypoczD7_dZ940yJzzAUV-SED2DOrl2BNEljwu5pmNtbZQlnNY_l29Mo91yOoU/w320-h216/kongruen10.png\" title=\"Congruence by ASA\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdRI1oBtJiRc5ROsGFOdUXCbyxMRGQ89OnC2q8fDMU8aEXlpGaenlVPDSqW9uiRdPHnvyun-BJGZelDfypoczD7_dZ940yJzzAUV-SED2DOrl2BNEljwu5pmNtbZQlnNY_l29Mo91yOoU/w320-h216/kongruen10.png\" data-sub-html=\"\u003ch2\u003eCongruence by ASA\u003c/h2\u003e\u003cp\u003eCongruence by ASA\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdRI1oBtJiRc5ROsGFOdUXCbyxMRGQ89OnC2q8fDMU8aEXlpGaenlVPDSqW9uiRdPHnvyun-BJGZelDfypoczD7_dZ940yJzzAUV-SED2DOrl2BNEljwu5pmNtbZQlnNY_l29Mo91yOoU/w320-h216/kongruen10.png' alt=\"Congruence by ASA\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCongruence by ASA\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdRI1oBtJiRc5ROsGFOdUXCbyxMRGQ89OnC2q8fDMU8aEXlpGaenlVPDSqW9uiRdPHnvyun-BJGZelDfypoczD7_dZ940yJzzAUV-SED2DOrl2BNEljwu5pmNtbZQlnNY_l29Mo91yOoU/s425/kongruen10.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdRI1oBtJiRc5ROsGFOdUXCbyxMRGQ89OnC2q8fDMU8aEXlpGaenlVPDSqW9uiRdPHnvyun-BJGZelDfypoczD7_dZ940yJzzAUV-SED2DOrl2BNEljwu5pmNtbZQlnNY_l29Mo91yOoU/s425/kongruen10.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nAB=DE,\\quad \\angle A=\\angle D,\\quad \\angle B=\\angle E\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8867em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle ABC \\cong \\triangle DEF\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example-in-a-kite\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-in-a-kite\" class=\"header-mark\" aria-label=\"Direct link to section: Example in a Kite\"\u003e\u003c/a\u003eExample in a Kite\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLM0hBvOROOw3Ae1pqvEdSlHeSme9n2U5bm_Dduey6pLp1Q7BNYQF_9Rusp1ocPqlqmLwljEqaP88rBB7T-Wx7GqhHWk0XWnG4urc4cE8Swidt6yivAU6akCcBRiWbGuQCD4UaBtc939U/w192-h200/kongruen12.png\" title=\"Congruent triangles in a kite\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLM0hBvOROOw3Ae1pqvEdSlHeSme9n2U5bm_Dduey6pLp1Q7BNYQF_9Rusp1ocPqlqmLwljEqaP88rBB7T-Wx7GqhHWk0XWnG4urc4cE8Swidt6yivAU6akCcBRiWbGuQCD4UaBtc939U/w192-h200/kongruen12.png\" data-sub-html=\"\u003ch2\u003eCongruent triangles in a kite\u003c/h2\u003e\u003cp\u003eCongruent triangles in a kite\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLM0hBvOROOw3Ae1pqvEdSlHeSme9n2U5bm_Dduey6pLp1Q7BNYQF_9Rusp1ocPqlqmLwljEqaP88rBB7T-Wx7GqhHWk0XWnG4urc4cE8Swidt6yivAU6akCcBRiWbGuQCD4UaBtc939U/w192-h200/kongruen12.png' alt=\"Congruent triangles in a kite\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCongruent triangles in a kite\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLM0hBvOROOw3Ae1pqvEdSlHeSme9n2U5bm_Dduey6pLp1Q7BNYQF_9Rusp1ocPqlqmLwljEqaP88rBB7T-Wx7GqhHWk0XWnG4urc4cE8Swidt6yivAU6akCcBRiWbGuQCD4UaBtc939U/s334/kongruen12.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLM0hBvOROOw3Ae1pqvEdSlHeSme9n2U5bm_Dduey6pLp1Q7BNYQF_9Rusp1ocPqlqmLwljEqaP88rBB7T-Wx7GqhHWk0XWnG4urc4cE8Swidt6yivAU6akCcBRiWbGuQCD4UaBtc939U/s334/kongruen12.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eSome congruent pairs that can be identified are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle AED \\cong \\triangle ABE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eBE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle DEC \\cong \\triangle BEC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eEC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBEC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle ACD \\cong \\triangle ABC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"d-finding-sides-and-angles-in-congruent-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-finding-sides-and-angles-in-congruent-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: D. Finding Sides and Angles in Congruent Triangles\"\u003e\u003c/a\u003eD. Finding Sides and Angles in Congruent Triangles\u003c/h2\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPAegiuVDnz6SJiq1LeKs66CLxI3QW9ZCHWZtJtWZLMIDzSdQWmqsd1yLW_q9LMC2q9juWguxnnY85CTxr082yvR-G7PbqybRmFdMMBeCItjXL-3jBpKBUSAG1aUhyphenhyphenWb_tABxsz3gneE/w200-h160/kongruen13.png\" title=\"Finding sides and angles in congruent triangles\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPAegiuVDnz6SJiq1LeKs66CLxI3QW9ZCHWZtJtWZLMIDzSdQWmqsd1yLW_q9LMC2q9juWguxnnY85CTxr082yvR-G7PbqybRmFdMMBeCItjXL-3jBpKBUSAG1aUhyphenhyphenWb_tABxsz3gneE/w200-h160/kongruen13.png\" data-sub-html=\"\u003ch2\u003eFinding sides and angles in congruent triangles\u003c/h2\u003e\u003cp\u003eFinding sides and angles in congruent triangles\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPAegiuVDnz6SJiq1LeKs66CLxI3QW9ZCHWZtJtWZLMIDzSdQWmqsd1yLW_q9LMC2q9juWguxnnY85CTxr082yvR-G7PbqybRmFdMMBeCItjXL-3jBpKBUSAG1aUhyphenhyphenWb_tABxsz3gneE/w200-h160/kongruen13.png' alt=\"Finding sides and angles in congruent triangles\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eFinding sides and angles in congruent triangles\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPAegiuVDnz6SJiq1LeKs66CLxI3QW9ZCHWZtJtWZLMIDzSdQWmqsd1yLW_q9LMC2q9juWguxnnY85CTxr082yvR-G7PbqybRmFdMMBeCItjXL-3jBpKBUSAG1aUhyphenhyphenWb_tABxsz3gneE/s310/kongruen13.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPAegiuVDnz6SJiq1LeKs66CLxI3QW9ZCHWZtJtWZLMIDzSdQWmqsd1yLW_q9LMC2q9juWguxnnY85CTxr082yvR-G7PbqybRmFdMMBeCItjXL-3jBpKBUSAG1aUhyphenhyphenWb_tABxsz3gneE/s310/kongruen13.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\triangle KNM \\cong \\triangle LNM\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eNM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eNM\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then the corresponding sides are equal:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nKM=ML=10\\text{ cm},\\quad KN=LN=5\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe length of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eMN\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eMN\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e75\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nMN=\\sqrt{10^2-5^2}=\\sqrt{75}=5\\sqrt{3}\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eMN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan 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style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle 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class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eand\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e∠\u003c/mi\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\angle KMN=\\angle NML=30^\\circ\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eMN\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6922em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e∠\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eNM\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7241em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"e-geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#e-geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: E. GeoGebra Simulation\"\u003e\u003c/a\u003eE. GeoGebra Simulation\u003c/h2\u003e\u003cp\u003eMove the points in the simulation below to observe how similar and congruent triangles behave.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/rmb3ucnf/width/600/height/400/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Kesebangunan dan Kekongruenan\" width=\"600px\"\u003e \u003c/iframe\u003e\n\u003cp\u003eNotice that:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e∼\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle ABC \\sim \\triangle DEF\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∼\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eand\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e≅\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e△\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\triangle ABC \\cong \\triangle A\u0026#x27;B\u0026#x27;C\u0026#x27;\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9963em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e△\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"f-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#f-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: F. Evaluation\"\u003e\u003c/a\u003eF. Evaluation\u003c/h2\u003e\u003cp\u003eAfter studying this material, try the following questions.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eA person stands 2 meters from a lamppost and casts a 3-meter shadow. If the person\u0026rsquo;s height is 1.8 meters, what is the height of the lamppost?\u003c/li\u003e\n\u003cli\u003eA photo measuring 12 cm x 15 cm is placed on cardboard. The top, left, and right margins are each 2 cm. If the photo and the cardboard are similar, what is the area of the cardboard?\u003c/li\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAD:DB=2:1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eDE=6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, find \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAfvmUiwjiMsp67EffHapOXv72iMba6cuCCavTUvRCNO2qcHyJ4vM1L0f07BVXwEy68MnawhwO9yfs6TpSbq6o8cgqdZgVRNVk8EUfOEpxZS3XnWwc9VUn6hzdmXpC-tLRwbJPCuMDt7g/w200-h153/kongruen15.png\" title=\"Similarity evaluation problem\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAfvmUiwjiMsp67EffHapOXv72iMba6cuCCavTUvRCNO2qcHyJ4vM1L0f07BVXwEy68MnawhwO9yfs6TpSbq6o8cgqdZgVRNVk8EUfOEpxZS3XnWwc9VUn6hzdmXpC-tLRwbJPCuMDt7g/w200-h153/kongruen15.png\" data-sub-html=\"\u003ch2\u003eSimilarity evaluation problem\u003c/h2\u003e\u003cp\u003eSimilarity evaluation problem\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAfvmUiwjiMsp67EffHapOXv72iMba6cuCCavTUvRCNO2qcHyJ4vM1L0f07BVXwEy68MnawhwO9yfs6TpSbq6o8cgqdZgVRNVk8EUfOEpxZS3XnWwc9VUn6hzdmXpC-tLRwbJPCuMDt7g/w200-h153/kongruen15.png' alt=\"Similarity evaluation problem\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eSimilarity evaluation problem\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAfvmUiwjiMsp67EffHapOXv72iMba6cuCCavTUvRCNO2qcHyJ4vM1L0f07BVXwEy68MnawhwO9yfs6TpSbq6o8cgqdZgVRNVk8EUfOEpxZS3XnWwc9VUn6hzdmXpC-tLRwbJPCuMDt7g/s298/kongruen15.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAfvmUiwjiMsp67EffHapOXv72iMba6cuCCavTUvRCNO2qcHyJ4vM1L0f07BVXwEy68MnawhwO9yfs6TpSbq6o8cgqdZgVRNVk8EUfOEpxZS3XnWwc9VUn6hzdmXpC-tLRwbJPCuMDt7g/s298/kongruen15.png\u003c/a\u003e)\u003c/p\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eIn a trapezoid with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e∥\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB\\parallel CD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmo\u003e∥\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eEF\\parallel AB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, if \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eDE:EA=5:3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eCD=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB=10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, find \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eEF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eEF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3yrIjC9oDrg3E2EQtArm8SXf0_4lgWgB0qU_1GQOHZBtPV0cMEb2j_o4wCmO8Ea6_Fgv20eG4aBN0QUjABbXbbX3U5T908T3upTJ8OcbtnFHv8D2XM_UWxQ432pcStFcD8boyWzPdnTg/w200-h173/kongruen16.png\" title=\"Trapezoid similarity evaluation problem\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3yrIjC9oDrg3E2EQtArm8SXf0_4lgWgB0qU_1GQOHZBtPV0cMEb2j_o4wCmO8Ea6_Fgv20eG4aBN0QUjABbXbbX3U5T908T3upTJ8OcbtnFHv8D2XM_UWxQ432pcStFcD8boyWzPdnTg/w200-h173/kongruen16.png\" data-sub-html=\"\u003ch2\u003eTrapezoid similarity evaluation problem\u003c/h2\u003e\u003cp\u003eTrapezoid similarity evaluation problem\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3yrIjC9oDrg3E2EQtArm8SXf0_4lgWgB0qU_1GQOHZBtPV0cMEb2j_o4wCmO8Ea6_Fgv20eG4aBN0QUjABbXbbX3U5T908T3upTJ8OcbtnFHv8D2XM_UWxQ432pcStFcD8boyWzPdnTg/w200-h173/kongruen16.png' alt=\"Trapezoid similarity evaluation problem\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eTrapezoid similarity evaluation problem\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3yrIjC9oDrg3E2EQtArm8SXf0_4lgWgB0qU_1GQOHZBtPV0cMEb2j_o4wCmO8Ea6_Fgv20eG4aBN0QUjABbXbbX3U5T908T3upTJ8OcbtnFHv8D2XM_UWxQ432pcStFcD8boyWzPdnTg/s353/kongruen16.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3yrIjC9oDrg3E2EQtArm8SXf0_4lgWgB0qU_1GQOHZBtPV0cMEb2j_o4wCmO8Ea6_Fgv20eG4aBN0QUjABbXbbX3U5T908T3upTJ8OcbtnFHv8D2XM_UWxQ432pcStFcD8boyWzPdnTg/s353/kongruen16.png\u003c/a\u003e)\u003c/p\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB=CD=8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAD=20\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, find \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg38SxRpYzmBStVr5LO46cwnmiS3zQVfXOjDmkiR9-0I1119TzNKqHlw7cqOFsJrYZ9GF_IOAAUgNbmEN1y2mVDYwS6_6jq-ZvRt4ZXttidu5S90PAAinoAQHRvVE7G7Z2dg8BRVKlkpiQ/w200-h171/kongruen17.png\" title=\"Two similar triangles evaluation problem\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg38SxRpYzmBStVr5LO46cwnmiS3zQVfXOjDmkiR9-0I1119TzNKqHlw7cqOFsJrYZ9GF_IOAAUgNbmEN1y2mVDYwS6_6jq-ZvRt4ZXttidu5S90PAAinoAQHRvVE7G7Z2dg8BRVKlkpiQ/w200-h171/kongruen17.png\" data-sub-html=\"\u003ch2\u003eTwo similar triangles evaluation problem\u003c/h2\u003e\u003cp\u003eTwo similar triangles evaluation problem\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg38SxRpYzmBStVr5LO46cwnmiS3zQVfXOjDmkiR9-0I1119TzNKqHlw7cqOFsJrYZ9GF_IOAAUgNbmEN1y2mVDYwS6_6jq-ZvRt4ZXttidu5S90PAAinoAQHRvVE7G7Z2dg8BRVKlkpiQ/w200-h171/kongruen17.png' alt=\"Two similar triangles evaluation problem\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eTwo similar triangles evaluation problem\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg38SxRpYzmBStVr5LO46cwnmiS3zQVfXOjDmkiR9-0I1119TzNKqHlw7cqOFsJrYZ9GF_IOAAUgNbmEN1y2mVDYwS6_6jq-ZvRt4ZXttidu5S90PAAinoAQHRvVE7G7Z2dg8BRVKlkpiQ/s317/kongruen17.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg38SxRpYzmBStVr5LO46cwnmiS3zQVfXOjDmkiR9-0I1119TzNKqHlw7cqOFsJrYZ9GF_IOAAUgNbmEN1y2mVDYwS6_6jq-ZvRt4ZXttidu5S90PAAinoAQHRvVE7G7Z2dg8BRVKlkpiQ/s317/kongruen17.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eThis material can be used as a foundation for solving many geometry problems involving ratios, proportions, and triangle properties.\u003c/p\u003e\n", "language": "en" }, { "title" : "Circle", "date_published" : "2021-01-24T08:30:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/unsur-unsur-lingkaran/", "url" : "https://www.aantriono.com/en/unsur-unsur-lingkaran/", "summary": " IntroductionThis lesson covers the definition of a circle, the elements of a circle, the approximation of π\\piπ, the formulas for circumference and area, and a few contextual examples.\nBasic Competencies3.7 Explain central angles, inscribed angles, arc length, and the area of a sector, along with their relationships.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eThis lesson covers the definition of a circle, the elements of a circle, the approximation of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the formulas for circumference and area, and a few contextual examples.\u003c/p\u003e\n\u003ch2 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h2\u003e\u003cp\u003e3.7 Explain central angles, inscribed angles, arc length, and the area of a sector, along with their relationships.\u003c/p\u003e\n\u003cp\u003e4.7 Solve problems related to central angles, inscribed angles, arc length, and the area of a sector.\u003c/p\u003e\n\u003ch2 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying this material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eDerive the formula for the circumference of a circle from contextual situations.\u003c/li\u003e\n\u003cli\u003eDerive the formula for the area of a circle from contextual situations.\u003c/li\u003e\n\u003cli\u003eExplain the relationship between a central angle and arc length.\u003c/li\u003e\n\u003cli\u003eExplain the relationship between a central angle and sector area.\u003c/li\u003e\n\u003cli\u003eSolve problems involving circumference and area of circles.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h2\u003e\u003ch3 id=\"1-definition-of-a-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-definition-of-a-circle\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Definition of a Circle\"\u003e\u003c/a\u003e1. Definition of a Circle\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikpRoQL9jyGMxjH4fgEAqKnYOSmelaScFCThEyh5TuMFfXEpkUYRL4L1iFZ8i2xayGO1kjlkQMw8UMarztGq5afg0csO0yewc04k0P7I9WABmEfoxSny4OpiDOygeUizzCoehrk-MZHwfFu5MLsJrbLS-mtIyq44NGyKxrxFGRLcJVH5z3omsyPKOJ/w640-h200/lingkar.png' alt=\"Definition of a circle\" \u003e\u003c/p\u003e\n\u003cp\u003eA circle is a simple closed curve whose points are all the same distance from one fixed point. That fixed point is called the center of the circle. The distance from the center to a point on the circle is called the radius.\u003c/p\u003e\n\u003cp\u003eThe region enclosed by the circle is called the circular region.\u003c/p\u003e\n\u003ch3 id=\"2-elements-of-a-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-elements-of-a-circle\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Elements of a Circle\"\u003e\u003c/a\u003e2. Elements of a Circle\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNHtwQpuX6GRfnZD8rwbp0zXqR3cO5BHlz-CT4XV7TCYjlR6noMeo76ShLbtJJDSLbA0uTxQay3dRSQjk7Cxv8IF7okZS41yxpGU2ENTbMok2tnY-nZXvv98YrN6IWR05V8_HaD5kM9sU/w320-h233/lingkaran2.png' alt=\"Elements of a circle\" \u003e\u003c/p\u003e\n\u003cp\u003eImportant elements of a circle include:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eCenter\u003c/strong\u003e, the point equidistant from all points on the circle, for example point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eO\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eRadius\u003c/strong\u003e, a line segment from the center to a point on the circle, for example \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{OA}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{OB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eOB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{OC}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eOC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eDiameter\u003c/strong\u003e, a line segment connecting two points on the circle and passing through the center, for example \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eArc\u003c/strong\u003e, a curved part of the circle, for example arc \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e^\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\widehat{AB}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord accent\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"svg-align\" style=\"top:-3.6833em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan style=\"height:0.3em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.3em\" viewBox=\"0 0 2364 300\" preserveAspectRatio=\"none\"\u003e\u003cpath d=\"M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10\n-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eChord\u003c/strong\u003e, a line segment connecting two points on the circle, for example \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{AC}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSector\u003c/strong\u003e, the region bounded by two radii and an arc, for example sector \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBOC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBOC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eSegment\u003c/strong\u003e, the region bounded by a chord and its arc, for example segment \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eADC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eApothem\u003c/strong\u003e, the shortest segment from the center to a chord, for example \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e‾\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\overline{OE}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord overline\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8833em;\"\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eOE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8033em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"3-approximating-the-value-of\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-approximating-the-value-of\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Approximating the Value of π\\piπ\"\u003e\u003c/a\u003e3. Approximating the Value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYmotHX_bXYOvBbJHcvQVp1xqKoolatrn3-s88bA0TVtGoG5X_OhN651KlfNMJAgRBS122YQM5da6QMpU1i3RDwULS8DLVTq4MKTttydK82z8AdXygevRZ0PWjfRHCvJDTm9vE3ArqjD4/w320-h297/lingkaran3.png' alt=\"Circumference of a circle\" \u003e\u003c/p\u003e\n\u003cp\u003eThe circumference of a circle is the length of the path around the edge of the circle.\u003c/p\u003e\n\u003cp\u003eTo approximate \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, try the following activity:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eFind several circular objects such as coins, wheels, or cans.\u003c/li\u003e\n\u003cli\u003eMeasure the circumference and the diameter of each object.\u003c/li\u003e\n\u003cli\u003eCompute \u003ccode\u003ecircumference / diameter\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eRecord the results in a table.\u003c/li\u003e\n\u003c/ol\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003eNo\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eObject\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eCircumference\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eDiameter\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eCircumference / Diameter\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eFirst object\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eSecond object\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003eThird object\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u0026hellip;\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eWith careful measurement, the ratio will approach a constant number, namely \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eCommon approximations are:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi \\approx 3.14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4831em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi \\approx \\frac{22}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4831em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"4-circumference-of-a-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-circumference-of-a-circle\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Circumference of a Circle\"\u003e\u003c/a\u003e4. Circumference of a Circle\u003c/h3\u003e\u003cp\u003eFrom the activity above:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmo\u003e⟺\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\pi = \\frac{K}{d}\n\\quad \\Longleftrightarrow \\quad\nK = \\pi d\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⟺\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed = 2r\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nK = \\pi d = 2\\pi r\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the circumference formula is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nK = \\pi d \\quad \\text{or} \\quad K = 2\\pi r\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eK\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = circumference\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = diameter\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = radius\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eUsually, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi = \\frac{22}{7}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is used when the radius or diameter is a multiple of 7, while \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi = 3.14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is used in other cases.\u003c/p\u003e\n\u003ch4 id=\"examples\" class=\"headerLink\"\u003e\n \u003ca href=\"#examples\" class=\"header-mark\" aria-label=\"Direct link to section: Examples\"\u003e\u003c/a\u003eExamples\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eFind the circumference of a circle with radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e21\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e21\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/li\u003e\n\u003cli\u003eA motorcycle wheel has radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e28\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e28\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e28\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\n\u003cul\u003e\n\u003cli\u003eFind the circumference.\u003c/li\u003e\n\u003cli\u003eFind the distance traveled after \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e100\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e rotations.\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003eA circular garden has diameter \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e105\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e105\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e105\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m. If palm trees are planted every \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e meters along the boundary, how many trees are needed?\u003c/li\u003e\n\u003cli\u003eFind the perimeter of the shaded region in the following two figures.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh94KbdpUiXG_Qbs-Td4EJvg8PnkMp-qAPke-saSH789039uemBN7UCdFttM6HcC4W9cLhWLbcZHJxT0U-actOoPg1Onz7knw2XFw7ED9bS0xnf3-bEwWQYMRMkFpf0fOrT_cbm_kbX_o8/w200-h179/lingkaran7.png' alt=\"Circumference example 1\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCudsL_-2OjevfW2NYKuDyzVJriFdgVi6drlSYyXKbhTa5zLNNVLkloEyJ9IvNE_5A6t3RxuJWi8But771DMJFwQVK7PtpLjl5WPVH8o79ewbw35bRQKNbRPDZA7f1qon5XeFlvL5RUH0/w200-h138/lingkaran10.png' alt=\"Circumference example 2\" \u003e\u003c/p\u003e\n\u003ch4 id=\"solutions\" class=\"headerLink\"\u003e\n \u003ca href=\"#solutions\" class=\"header-mark\" aria-label=\"Direct link to section: Solutions\"\u003e\u003c/a\u003eSolutions\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e132\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nK = 2\\pi r = 2 \\times \\frac{22}{7} \\times 21 = 132\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e132\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the circumference is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e132\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e132\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e132\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e28\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e176\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nK = 2\\pi r = 2 \\times \\frac{22}{7} \\times 28 = 176\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e28\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e176\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the circumference is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e176\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e176\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e176\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/p\u003e\n\u003cp\u003eDistance traveled after \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e100\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e rotations:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e176\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e17,600\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e176\u003c/mn\u003e\u003cmtext\u003e m\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n176 \\times 100 = 17{,}600 \\text{ cm} = 176 \\text{ m}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e176\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e17\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e600\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e176\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e m\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"3\"\u003e\n\u003cli\u003eGarden boundary:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e105\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e330\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nK \u0026amp;= \\pi d \\\\\n \u0026amp;= \\frac{22}{7} \\times 105 \\\\\n \u0026amp;= 330\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3074em;vertical-align:-2.4037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e105\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e330\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eNumber of trees:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e330\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e55\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nn \u0026amp;= \\frac{330}{6} \\\\\n \u0026amp;= 55\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.8074em;vertical-align:-1.6537em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.1537em;\"\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.6537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.1537em;\"\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e330\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e55\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.6537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e55\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e55\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e55\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e palm trees are needed.\u003c/p\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003eFirst figure:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e⊙\u003c/mo\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e⊙\u003c/mo\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e⊙\u003c/mo\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmo\u003e⊙\u003c/mo\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e82.8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nK \u0026amp;= \\frac{1}{2}K_\\odot + \\frac{1}{4}K_\\odot + \\frac{1}{4}K_\\odot + 2r \\\\\n \u0026amp;= K_\\odot + 2r \\\\\n \u0026amp;= 2\\pi r + 2r \\\\\n \u0026amp;= 2 \\times 3.14 \\times 10 + 2 \\times 10 \\\\\n \u0026amp;= 82.8\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:8.3074em;vertical-align:-3.9037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4037em;\"\u003e\u003cspan style=\"top:-6.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.9037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.4037em;\"\u003e\u003cspan style=\"top:-6.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2583em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e⊙\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2083em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2583em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e⊙\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2083em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2583em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e⊙\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2083em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e82.8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.9037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the perimeter of the shaded region is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e82.8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e82.8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e82.8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/p\u003e\n\u003cp\u003eSecond figure:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmtext\u003elarge\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmtext\u003esmall\u003c/mtext\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmtext\u003elarge\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmtext\u003elarge\u003c/mtext\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eK\u003c/mi\u003e\u003cmtext\u003elarge\u003c/mtext\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e21\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e66\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nK \u0026amp;= \\frac{1}{2}K_{\\text{large}} + K_{\\text{small}} \\\\\n \u0026amp;= \\frac{1}{2}K_{\\text{large}} + \\frac{1}{2}K_{\\text{large}} \\\\\n \u0026amp;= K_{\\text{large}} \\\\\n \u0026amp;= \\frac{22}{7} \\times 21 \\\\\n \u0026amp;= 66\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:9.9223em;vertical-align:-4.7112em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.2112em;\"\u003e\u003cspan style=\"top:-7.2112em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.0963em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.7297em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7112em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.2112em;\"\u003e\u003cspan style=\"top:-7.2112em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003elarge\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003esmall\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003elarge\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003elarge\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eK\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003elarge\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.0963em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.7297em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e66\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7112em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the perimeter of the shaded region is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e66\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e66\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e66\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/p\u003e\n\u003ch3 id=\"5-area-of-a-circle\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-area-of-a-circle\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Area of a Circle\"\u003e\u003c/a\u003e5. Area of a Circle\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn70RMt5epJVytmtbl-c7uk433c01pcXCSRhi92QD2Zcmvj3UO-NTNaVwRhVpZZt3C5L3DdNPjdsbcpsSS7H2M5FkeZ4ky8BAdYBhMIxnHQr2mhwZhB9WIMGEiDOrqYGnkcQyOu5mCX8w/w320-h297/lingkaran14.png' alt=\"Area of a circle\" \u003e\u003c/p\u003e\n\u003cp\u003eThe area of a circle is the area enclosed by its circumference.\u003c/p\u003e\n\u003cp\u003eOne way to derive the formula is:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eDraw a circle.\u003c/li\u003e\n\u003cli\u003eDivide it into two equal parts.\u003c/li\u003e\n\u003cli\u003eSplit it further into several sectors.\u003c/li\u003e\n\u003cli\u003eRearrange the sectors so that the shape approaches a rectangle.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhq4NDFNTgkyqtAlUxDTlBCAQbqoE6JkDfpG1YZ3XXEd23Z-5VmnjcvhFC6TS5_ZZKKv9uN4uA1xAM2Z6hBwqK1o2BR7ZDHXFH2VWlFkvtzOkZk3_SRBv5Aq6LUXhwckMle1mKXoCS_zs/w400-h181/lingkaran4.png' alt=\"Deriving the area formula\" \u003e\u003c/p\u003e\n\u003cp\u003eAs the number of sectors increases, the new shape approaches a rectangle with:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003elength \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e= \\frac{1}{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.3669em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e circumference\u003c/li\u003e\n\u003cli\u003ewidth \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e= r\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.3669em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThus:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003el\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003ecircumference\u003c/mtext\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= p \\times l \\\\\n \u0026amp;= \\frac{1}{2} \\times \\text{circumference} \\times r \\\\\n \u0026amp;= \\frac{1}{2} \\times 2\\pi r \\times r \\\\\n \u0026amp;= \\pi r^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.639em;vertical-align:-3.5695em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0695em;\"\u003e\u003cspan style=\"top:-6.5509em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5695em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.2621em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.4119em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5695em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0695em;\"\u003e\u003cspan style=\"top:-6.5509em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.01968em;\"\u003el\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5695em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecircumference\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.2621em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.4119em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5695em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIn terms of diameter:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= \\pi r^2 \\\\\n \u0026amp;= \\pi \\left(\\frac{d}{2}\\right)^2 \\\\\n \u0026amp;= \\frac{1}{4}\\pi d^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.7356em;vertical-align:-3.1178em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6178em;\"\u003e\u003cspan style=\"top:-6.4077em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.0937em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5222em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1178em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6178em;\"\u003e\u003cspan style=\"top:-6.4077em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.0937em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5222em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1178em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the area formula is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eor\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL = \\pi r^2\n\\quad \\text{or} \\quad\nL = \\frac{1}{4}\\pi d^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eor\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = area\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = radius\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = diameter\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"examples-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#examples-1\" class=\"header-mark\" aria-label=\"Direct link to section: Examples\"\u003e\u003c/a\u003eExamples\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eFind the area of a circle if:\n\u003cul\u003e\n\u003cli\u003ethe radius is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm\u003c/li\u003e\n\u003cli\u003ethe diameter is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003eFind the radius if the area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e154\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e154\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e154\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eFind the area of the shaded region in the following three figures.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqF3O9Gt4on5xaUjMmZBnzACwEIsfJWmiayd94zppLOFb6hK9slIz113XKL0sp9TzlKP8LIDcuHlpQsgeDlAPLTRx-Ij5tfgx3Cg2slmtT-3VupuQJZ1jUvGJdkQGK0NBKX_Msg6kG_IY/s320/lingkaran9.png' alt=\"Area example 1\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtjZbCJQbgFlBYg8qvk4t4jEfahzORh3CaCLOl0cskydg_7MQLGV9-SEHFKr7LvRs9_y2CvdVV2w-I7q9mWjPay02kIqZsck0xUw5XkVumn4z3p_1tj7II2VOfSC1TNA-3PjKvqTITiSA/s231/lingkaran12.png' alt=\"Area example 2\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyr5RNqHX1ztCWo2-8HLHc1isEX16Obh9CLLWGpZIAP2VS_Enm7oQx31W_s-nE9w2tV93hsElm8-hQ94S8isMOb902jZ4-8ILJK-6JSIW7z8wdukyE1X-Jey0G7ZEn4ViKSKjoyEBj9sw/s303/lingkaran8.png' alt=\"Area example 3\" \u003e\u003c/p\u003e\n\u003ch4 id=\"solutions-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#solutions-1\" class=\"header-mark\" aria-label=\"Direct link to section: Solutions\"\u003e\u003c/a\u003eSolutions\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er = 10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e314\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= \\pi r^2 \\\\\n \u0026amp;= 3.14 \\times 10^2 \\\\\n \u0026amp;= 314\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e314\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e314\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e314\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e314\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed = 7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e38.5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= \\frac{1}{4}\\pi d^2 \\\\\n \u0026amp;= \\frac{1}{4} \\times \\frac{22}{7} \\times 7^2 \\\\\n \u0026amp;= 38.5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.1149em;vertical-align:-2.8074em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.3074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.3074em;\"\u003e\u003cspan style=\"top:-5.3074em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e38.5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.8074em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e38.5\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e38.5\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e38.5\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e154\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e49\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n154 \u0026amp;= \\frac{22}{7}r^2 \\\\\nr^2 \u0026amp;= 49 \\\\\nr \u0026amp;= 7\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3315em;vertical-align:-2.4158em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9158em;\"\u003e\u003cspan style=\"top:-4.9158em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e154\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4158em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9158em;\"\u003e\u003cspan style=\"top:-4.9158em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e49\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4158em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the radius is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003eFirst figure:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eArea of the large square:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e196\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n14 \\times 14 = 196\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e196\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eShaded area:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e49\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e77\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e126\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= 7 \\times 7 + 2 \\times \\frac{22}{7} \\times \\left(\\frac{7}{2}\\right)^2 \\\\\n \u0026amp;= 49 + 77 \\\\\n \u0026amp;= 126\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.904em;vertical-align:-2.702em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.202em;\"\u003e\u003cspan style=\"top:-5.202em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.112em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.612em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.702em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.202em;\"\u003e\u003cspan style=\"top:-5.202em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.112em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e49\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e77\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.612em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.654em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e126\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.702em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eUnshaded area:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e196\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e126\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e70\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n196 - 126 = 70\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e196\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e126\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e70\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the unshaded area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e70\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e70\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e70\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eSecond figure:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e49\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e38.5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10.5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL \u0026amp;= 7 \\times 7 - \\frac{1}{4} \\times \\frac{22}{7} \\times 7^2 \\\\\n \u0026amp;= 49 - 38.5 \\\\\n \u0026amp;= 10.5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3074em;vertical-align:-2.4037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-4.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-4.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e49\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e38.5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10.5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the shaded area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10.5\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10.5\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10.5\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThird figure:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmtext\u003esegment\u003c/mtext\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e154\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e98\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e56\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL_{\\text{segment}} \u0026amp;= \\frac{1}{4} \\times \\frac{22}{7} \\times 14^2 - \\frac{1}{2} \\times 14 \\times 14 \\\\\n \u0026amp;= 154 - 98 \\\\\n \u0026amp;= 56\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3074em;vertical-align:-2.4037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-4.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2806em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003esegment\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-4.9037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e154\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e98\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e56\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince there are two equal segments:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmtext\u003etotal\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e56\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e112\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL_{\\text{total}} = 2 \\times 56 = 112\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003etotal\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e56\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e112\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the total shaded area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e112\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e112\\text{ cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e112\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"6-circle-optical-illusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-circle-optical-illusion\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Circle Optical Illusion\"\u003e\u003c/a\u003e6. Circle Optical Illusion\u003c/h3\u003e\u003cp\u003eThe following image is a circle optical illusion previously created using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmtext\u003eLaTeX\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\LaTeX\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8988em;vertical-align:-0.2155em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.36em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6833em;\"\u003e\u003cspan style=\"top:-2.905em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm mtight sizing reset-size6 size3\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord textrm\"\u003eT\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4678em;\"\u003e\u003cspan style=\"top:-2.7845em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord textrm\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2155em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:-0.125em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord textrm\"\u003eX\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/a/AVvXsEhaJq8vQJ04uEnbhGSlApqW6GYIJv2vNQTbsU0eB6OduuCNX1V4XZihfHetaI4nAAKhQFLv_c5y6FZOZsXwDTrDZqCjoPjtNg7M9nKMFNFjYaAW5XGQH7XV4zI_Q1axerg4DWS-Qpo2l3RABg52ObuGQXLcE3QZy4oai5H2iU1Wbk1rUS_5r6qPLj7d=w320-h159' alt=\"Circle optical illusion\" \u003e\u003c/p\u003e\n\u003ch3 id=\"7-geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: 7. GeoGebra Simulation\"\u003e\u003c/a\u003e7. GeoGebra Simulation\u003c/h3\u003e\u003cp\u003eThis simulation shows the parts of a circle. Turn the elements on or off to inspect them.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/ybddpwwp/width/600/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/true/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Bagian-bagian lingkaran\" width=\"600px\"\u003e\u003c/iframe\u003e\n\u003ch3 id=\"8-circumference-and-area-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#8-circumference-and-area-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: 8. Circumference and Area Simulation\"\u003e\u003c/a\u003e8. Circumference and Area Simulation\u003c/h3\u003e\u003cp\u003eMove the radius slider to see how the circumference and area change.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/spzf7pxv/width/600/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Keliling dan Luas Lingkaran\" width=\"600px\"\u003e\u003c/iframe\u003e\n\u003ch3 id=\"9-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#9-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: 9. Evaluation\"\u003e\u003c/a\u003e9. Evaluation\u003c/h3\u003e\u003cp\u003eThe old interactive worksheet in this article depended on a third-party embed that is no longer stable with the current theme. It has been removed to keep the page rendering intact.\u003c/p\u003e\n\u003cp\u003eIf needed, the evaluation can be replaced later with:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ea stable Hugo shortcode,\u003c/li\u003e\n\u003cli\u003ean external worksheet link,\u003c/li\u003e\n\u003cli\u003eor a local quiz component.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThis completes the circle lesson and its main elements. Suggestions and corrections can be added through the available feedback channel.\u003c/p\u003e\n", "language": "en" }, { "title" : "Cone", "date_published" : "2021-01-16T16:25:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kerucut/", "url" : "https://www.aantriono.com/en/kerucut/", "summary": " IntroductionA cone is a solid with one circular base and one apex. This lesson covers the elements of a cone, its slant height, its net, surface area, volume, and several common example problems.\nA. Basic Competencies3.7 Generalize the surface area and volume of curved solids, especially cylinders, cones, and spheres.\n4.7 Solve contextual problems involving the surface area and volume of curved solids and combinations of those solids.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eA cone is a solid with one circular base and one apex. This lesson covers the elements of a cone, its slant height, its net, surface area, volume, and several common example problems.\u003c/p\u003e\n\u003ch2 id=\"a-basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: A. Basic Competencies\"\u003e\u003c/a\u003eA. Basic Competencies\u003c/h2\u003e\u003cp\u003e3.7 Generalize the surface area and volume of curved solids, especially cylinders, cones, and spheres.\u003cbr\u003e\n4.7 Solve contextual problems involving the surface area and volume of curved solids and combinations of those solids.\u003c/p\u003e\n\u003ch2 id=\"b-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: B. Learning Objectives\"\u003e\u003c/a\u003eB. Learning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying this material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eidentify the elements of a cone,\u003c/li\u003e\n\u003cli\u003ecalculate the base area, lateral area, and total surface area,\u003c/li\u003e\n\u003cli\u003ecalculate the volume of a cone,\u003c/li\u003e\n\u003cli\u003edetermine unknown cone elements from given data,\u003c/li\u003e\n\u003cli\u003ecompare cone volumes after size changes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-material\" class=\"header-mark\" aria-label=\"Direct link to section: C. Material\"\u003e\u003c/a\u003eC. Material\u003c/h2\u003e\u003ch3 id=\"1-elements-of-a-cone\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-elements-of-a-cone\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Elements of a Cone\"\u003e\u003c/a\u003e1. Elements of a Cone\u003c/h3\u003e\u003cp\u003eLook at the figure below.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipRkyYJwojSRjEP4QDXqpWp-KRgJ_zyQ5WBrec-OVLQZEL0YDD4cwlWw6yRJcrv3VGrdIm6RtU9PIkVBcGz1rrJp6D9aR_LP-VsgP3r87kEdvTt21z6baSf9GuDCeT1pOpQqmbZWrYtJS5ELzxsIO9VAVqit3iZOY8s18rxmeYi9JmOEJSd1XaGiCI/w640-h200/kerucut.png\" title=\"Elements of a cone\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipRkyYJwojSRjEP4QDXqpWp-KRgJ_zyQ5WBrec-OVLQZEL0YDD4cwlWw6yRJcrv3VGrdIm6RtU9PIkVBcGz1rrJp6D9aR_LP-VsgP3r87kEdvTt21z6baSf9GuDCeT1pOpQqmbZWrYtJS5ELzxsIO9VAVqit3iZOY8s18rxmeYi9JmOEJSd1XaGiCI/w640-h200/kerucut.png\" data-sub-html=\"\u003ch2\u003eElements of a cone\u003c/h2\u003e\u003cp\u003eElements of a cone\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipRkyYJwojSRjEP4QDXqpWp-KRgJ_zyQ5WBrec-OVLQZEL0YDD4cwlWw6yRJcrv3VGrdIm6RtU9PIkVBcGz1rrJp6D9aR_LP-VsgP3r87kEdvTt21z6baSf9GuDCeT1pOpQqmbZWrYtJS5ELzxsIO9VAVqit3iZOY8s18rxmeYi9JmOEJSd1XaGiCI/w640-h200/kerucut.png' alt=\"Elements of a cone\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eElements of a cone\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipRkyYJwojSRjEP4QDXqpWp-KRgJ_zyQ5WBrec-OVLQZEL0YDD4cwlWw6yRJcrv3VGrdIm6RtU9PIkVBcGz1rrJp6D9aR_LP-VsgP3r87kEdvTt21z6baSf9GuDCeT1pOpQqmbZWrYtJS5ELzxsIO9VAVqit3iZOY8s18rxmeYi9JmOEJSd1XaGiCI/s1280/kerucut.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipRkyYJwojSRjEP4QDXqpWp-KRgJ_zyQ5WBrec-OVLQZEL0YDD4cwlWw6yRJcrv3VGrdIm6RtU9PIkVBcGz1rrJp6D9aR_LP-VsgP3r87kEdvTt21z6baSf9GuDCeT1pOpQqmbZWrYtJS5ELzxsIO9VAVqit3iZOY8s18rxmeYi9JmOEJSd1XaGiCI/s1280/kerucut.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eA cone has:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eone flat circular base,\u003c/li\u003e\n\u003cli\u003eone curved lateral surface,\u003c/li\u003e\n\u003cli\u003eone apex,\u003c/li\u003e\n\u003cli\u003ea base radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003ea height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003et\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003ea circular edge where the base meets the lateral surface.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-slant-height\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-slant-height\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Slant Height\"\u003e\u003c/a\u003e2. Slant Height\u003c/h3\u003e\u003cp\u003eConsider the right triangle below.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXDsuQZIn3fc4P9dU1QaRQy3WgYJGBNTI-yOKEFecoJiJGAB45fbTHLuUvM5y3aQITtv0nNS7exnGp6Iz3pF8MTbXrzM_VA3nicru9IsVwwtUbo9eOiEwdYQBm36HdKa6dy3QeXoA9qnY/w200-h181/kerucut1.png\" title=\"Triangle forming a cone\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXDsuQZIn3fc4P9dU1QaRQy3WgYJGBNTI-yOKEFecoJiJGAB45fbTHLuUvM5y3aQITtv0nNS7exnGp6Iz3pF8MTbXrzM_VA3nicru9IsVwwtUbo9eOiEwdYQBm36HdKa6dy3QeXoA9qnY/w200-h181/kerucut1.png\" data-sub-html=\"\u003ch2\u003eTriangle forming a cone\u003c/h2\u003e\u003cp\u003eTriangle forming a cone\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXDsuQZIn3fc4P9dU1QaRQy3WgYJGBNTI-yOKEFecoJiJGAB45fbTHLuUvM5y3aQITtv0nNS7exnGp6Iz3pF8MTbXrzM_VA3nicru9IsVwwtUbo9eOiEwdYQBm36HdKa6dy3QeXoA9qnY/w200-h181/kerucut1.png' alt=\"Triangle forming a cone\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eTriangle forming a cone\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXDsuQZIn3fc4P9dU1QaRQy3WgYJGBNTI-yOKEFecoJiJGAB45fbTHLuUvM5y3aQITtv0nNS7exnGp6Iz3pF8MTbXrzM_VA3nicru9IsVwwtUbo9eOiEwdYQBm36HdKa6dy3QeXoA9qnY/s253/kerucut1.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXDsuQZIn3fc4P9dU1QaRQy3WgYJGBNTI-yOKEFecoJiJGAB45fbTHLuUvM5y3aQITtv0nNS7exnGp6Iz3pF8MTbXrzM_VA3nicru9IsVwwtUbo9eOiEwdYQBm36HdKa6dy3QeXoA9qnY/s253/kerucut1.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eWhen the triangle is rotated around its height, it forms a cone.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQMmr8cibcUHh8FddJA9zq8GpNGMiQ_maOL6y0eETO71v5Gg8jLJbyYQpXeqOZmP80PdTdTVm1b0Pmv038WLfGGshTh94pVn-Ri3BnLlMjcvGaI6dKKc_u2Fc_j1ilrivI68q4YjXDONc/w320-h298/kerucut2.png\" title=\"Cone and its slant height\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQMmr8cibcUHh8FddJA9zq8GpNGMiQ_maOL6y0eETO71v5Gg8jLJbyYQpXeqOZmP80PdTdTVm1b0Pmv038WLfGGshTh94pVn-Ri3BnLlMjcvGaI6dKKc_u2Fc_j1ilrivI68q4YjXDONc/w320-h298/kerucut2.png\" data-sub-html=\"\u003ch2\u003eCone and its slant height\u003c/h2\u003e\u003cp\u003eCone and its slant height\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQMmr8cibcUHh8FddJA9zq8GpNGMiQ_maOL6y0eETO71v5Gg8jLJbyYQpXeqOZmP80PdTdTVm1b0Pmv038WLfGGshTh94pVn-Ri3BnLlMjcvGaI6dKKc_u2Fc_j1ilrivI68q4YjXDONc/w320-h298/kerucut2.png' alt=\"Cone and its slant height\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCone and its slant height\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQMmr8cibcUHh8FddJA9zq8GpNGMiQ_maOL6y0eETO71v5Gg8jLJbyYQpXeqOZmP80PdTdTVm1b0Pmv038WLfGGshTh94pVn-Ri3BnLlMjcvGaI6dKKc_u2Fc_j1ilrivI68q4YjXDONc/s310/kerucut2.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQMmr8cibcUHh8FddJA9zq8GpNGMiQ_maOL6y0eETO71v5Gg8jLJbyYQpXeqOZmP80PdTdTVm1b0Pmv038WLfGGshTh94pVn-Ri3BnLlMjcvGaI6dKKc_u2Fc_j1ilrivI68q4YjXDONc/s310/kerucut2.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eThe segment from the apex to a point on the edge of the base is called the \u003cstrong\u003eslant height\u003c/strong\u003e, denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003es\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eBy the Pythagorean theorem:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ns^2=t^2+r^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ns=\\sqrt{t^2+r^2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0223em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1777em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-cone-net\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-cone-net\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Cone Net\"\u003e\u003c/a\u003e3. Cone Net\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3qysAQ50IeUTAk7v0xUsixfxBI7w8Pj1mo0jOlLJ8DW0IsUBH_imp7gLqY0VXdpO2Ha9xzg_AsTOYg7q998pweVcYodmIs_V08kL6XkYIvU2PzpqpceLeY3bS4Z5va67sn7AQWAMKF6A/w320-h314/kerucut3.png\" title=\"Closed cone\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3qysAQ50IeUTAk7v0xUsixfxBI7w8Pj1mo0jOlLJ8DW0IsUBH_imp7gLqY0VXdpO2Ha9xzg_AsTOYg7q998pweVcYodmIs_V08kL6XkYIvU2PzpqpceLeY3bS4Z5va67sn7AQWAMKF6A/w320-h314/kerucut3.png\" data-sub-html=\"\u003ch2\u003eClosed cone\u003c/h2\u003e\u003cp\u003eClosed cone\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3qysAQ50IeUTAk7v0xUsixfxBI7w8Pj1mo0jOlLJ8DW0IsUBH_imp7gLqY0VXdpO2Ha9xzg_AsTOYg7q998pweVcYodmIs_V08kL6XkYIvU2PzpqpceLeY3bS4Z5va67sn7AQWAMKF6A/w320-h314/kerucut3.png' alt=\"Closed cone\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eClosed cone\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3qysAQ50IeUTAk7v0xUsixfxBI7w8Pj1mo0jOlLJ8DW0IsUBH_imp7gLqY0VXdpO2Ha9xzg_AsTOYg7q998pweVcYodmIs_V08kL6XkYIvU2PzpqpceLeY3bS4Z5va67sn7AQWAMKF6A/s307/kerucut3.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3qysAQ50IeUTAk7v0xUsixfxBI7w8Pj1mo0jOlLJ8DW0IsUBH_imp7gLqY0VXdpO2Ha9xzg_AsTOYg7q998pweVcYodmIs_V08kL6XkYIvU2PzpqpceLeY3bS4Z5va67sn7AQWAMKF6A/s307/kerucut3.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eIf the lateral surface is opened, the net looks like this.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhCobjqQli5HC7zHsPOc2rRSnWaD_hPte2gU-VQ_zhfZvM4PELTonrRplZUPokgleEL859I_cv5QioB02BwVPXmBSJxscIrvS6hiv9Q3tDhzn-o0gUW7hofav-UG2YjTljOM0oghn9xY/w320-h302/kerucut4.png\" title=\"Cone net\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhCobjqQli5HC7zHsPOc2rRSnWaD_hPte2gU-VQ_zhfZvM4PELTonrRplZUPokgleEL859I_cv5QioB02BwVPXmBSJxscIrvS6hiv9Q3tDhzn-o0gUW7hofav-UG2YjTljOM0oghn9xY/w320-h302/kerucut4.png\" data-sub-html=\"\u003ch2\u003eCone net\u003c/h2\u003e\u003cp\u003eCone net\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhCobjqQli5HC7zHsPOc2rRSnWaD_hPte2gU-VQ_zhfZvM4PELTonrRplZUPokgleEL859I_cv5QioB02BwVPXmBSJxscIrvS6hiv9Q3tDhzn-o0gUW7hofav-UG2YjTljOM0oghn9xY/w320-h302/kerucut4.png' alt=\"Cone net\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCone net\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhCobjqQli5HC7zHsPOc2rRSnWaD_hPte2gU-VQ_zhfZvM4PELTonrRplZUPokgleEL859I_cv5QioB02BwVPXmBSJxscIrvS6hiv9Q3tDhzn-o0gUW7hofav-UG2YjTljOM0oghn9xY/s337/kerucut4.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhCobjqQli5HC7zHsPOc2rRSnWaD_hPte2gU-VQ_zhfZvM4PELTonrRplZUPokgleEL859I_cv5QioB02BwVPXmBSJxscIrvS6hiv9Q3tDhzn-o0gUW7hofav-UG2YjTljOM0oghn9xY/s337/kerucut4.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eA cone net consists of:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eone circle of radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e,\u003c/li\u003e\n\u003cli\u003eone sector of a circle with radius \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003es\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and arc length \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\pi r\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"4-surface-area-of-a-cone\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-surface-area-of-a-cone\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Surface Area of a Cone\"\u003e\u003c/a\u003e4. Surface Area of a Cone\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCCqadpoQWCk7tgN2JkA0D3z0GGwRqXfsbjb3IHv7G5NPgzH_IBeHowzdZFwG1hRr89H3L7ueyBwOKx02g36aL4OLO0-u8qKPXbAI61e1oirc1mLBLxlQPPBYCvbVNCXyYOF0FdVxJHg0/w400-h221/kerucut5.png\" title=\"Cone for surface area\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCCqadpoQWCk7tgN2JkA0D3z0GGwRqXfsbjb3IHv7G5NPgzH_IBeHowzdZFwG1hRr89H3L7ueyBwOKx02g36aL4OLO0-u8qKPXbAI61e1oirc1mLBLxlQPPBYCvbVNCXyYOF0FdVxJHg0/w400-h221/kerucut5.png\" data-sub-html=\"\u003ch2\u003eCone for surface area\u003c/h2\u003e\u003cp\u003eCone for surface area\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCCqadpoQWCk7tgN2JkA0D3z0GGwRqXfsbjb3IHv7G5NPgzH_IBeHowzdZFwG1hRr89H3L7ueyBwOKx02g36aL4OLO0-u8qKPXbAI61e1oirc1mLBLxlQPPBYCvbVNCXyYOF0FdVxJHg0/w400-h221/kerucut5.png' alt=\"Cone for surface area\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCone for surface area\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCCqadpoQWCk7tgN2JkA0D3z0GGwRqXfsbjb3IHv7G5NPgzH_IBeHowzdZFwG1hRr89H3L7ueyBwOKx02g36aL4OLO0-u8qKPXbAI61e1oirc1mLBLxlQPPBYCvbVNCXyYOF0FdVxJHg0/s545/kerucut5.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCCqadpoQWCk7tgN2JkA0D3z0GGwRqXfsbjb3IHv7G5NPgzH_IBeHowzdZFwG1hRr89H3L7ueyBwOKx02g36aL4OLO0-u8qKPXbAI61e1oirc1mLBLxlQPPBYCvbVNCXyYOF0FdVxJHg0/s545/kerucut5.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eThe lateral area of a cone is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL_s=\\pi rs\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003es\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ers\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe total surface area is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL=\\pi r^2+\\pi rs\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ers\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL=\\pi r(r+s)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h4\u003e\u003cp\u003eA cone has \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er=6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003et=8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm. Find its surface area using \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi=3.14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eFirst find the slant height:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ns=\\sqrt{6^2+8^2}=\\sqrt{100}=10\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0223em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1777em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0839em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e301.44\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL=\\pi r(r+s)=3.14\\times 6\\times (6+10)=301.44\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e301.44\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the surface area is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e301.44\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n301.44\\text{ cm}^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e301.44\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h4\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er=7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003es=14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e462\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nL=\\pi r(r+s)=\\frac{22}{7}\\times 7\\times (7+14)=462\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e462\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the surface area is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e462\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n462\\text{ cm}^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e462\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"5-volume-of-a-cone\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-volume-of-a-cone\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Volume of a Cone\"\u003e\u003c/a\u003e5. Volume of a Cone\u003c/h3\u003e\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzglDu0ocqWjno7QKgpGHwKLDjNAgs221l2e1ICHEZez_b-QwlQ5JHxYCq5cuBmxxcjnD2ruUGFdJO_8qE-a0KRuXxzPi4r23ivOetNXw2W8wzd9uemTerwKA7tDfGCMN4LHd8UrKXIyY/w231-h320/kerucut8.png\" title=\"Cone as a limiting pyramid\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzglDu0ocqWjno7QKgpGHwKLDjNAgs221l2e1ICHEZez_b-QwlQ5JHxYCq5cuBmxxcjnD2ruUGFdJO_8qE-a0KRuXxzPi4r23ivOetNXw2W8wzd9uemTerwKA7tDfGCMN4LHd8UrKXIyY/w231-h320/kerucut8.png\" data-sub-html=\"\u003ch2\u003eCone as a limiting pyramid\u003c/h2\u003e\u003cp\u003eCone as a limiting pyramid\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzglDu0ocqWjno7QKgpGHwKLDjNAgs221l2e1ICHEZez_b-QwlQ5JHxYCq5cuBmxxcjnD2ruUGFdJO_8qE-a0KRuXxzPi4r23ivOetNXw2W8wzd9uemTerwKA7tDfGCMN4LHd8UrKXIyY/w231-h320/kerucut8.png' alt=\"Cone as a limiting pyramid\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCone as a limiting pyramid\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzglDu0ocqWjno7QKgpGHwKLDjNAgs221l2e1ICHEZez_b-QwlQ5JHxYCq5cuBmxxcjnD2ruUGFdJO_8qE-a0KRuXxzPi4r23ivOetNXw2W8wzd9uemTerwKA7tDfGCMN4LHd8UrKXIyY/s349/kerucut8.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzglDu0ocqWjno7QKgpGHwKLDjNAgs221l2e1ICHEZez_b-QwlQ5JHxYCq5cuBmxxcjnD2ruUGFdJO_8qE-a0KRuXxzPi4r23ivOetNXw2W8wzd9uemTerwKA7tDfGCMN4LHd8UrKXIyY/s349/kerucut8.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eA cone can be viewed as a limiting case of a pyramid with more and more base sides.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw4JhLC8tVJJ9p8eLC_3XqK-yQBHv95Th6kmt7OblScK-UY6BYZsKwptTQsfwaIaXwCBc6V2wGEHEVKp4wVAJbDzncg9wERCEgJW9q7ISZq_iSSY2Vpz9bu04Bfrs7mu2GRieV9iVyxdY/w270-h320/kerucut9.png\" title=\"Pyramid-to-cone idea\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw4JhLC8tVJJ9p8eLC_3XqK-yQBHv95Th6kmt7OblScK-UY6BYZsKwptTQsfwaIaXwCBc6V2wGEHEVKp4wVAJbDzncg9wERCEgJW9q7ISZq_iSSY2Vpz9bu04Bfrs7mu2GRieV9iVyxdY/w270-h320/kerucut9.png\" data-sub-html=\"\u003ch2\u003ePyramid-to-cone idea\u003c/h2\u003e\u003cp\u003ePyramid-to-cone idea\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw4JhLC8tVJJ9p8eLC_3XqK-yQBHv95Th6kmt7OblScK-UY6BYZsKwptTQsfwaIaXwCBc6V2wGEHEVKp4wVAJbDzncg9wERCEgJW9q7ISZq_iSSY2Vpz9bu04Bfrs7mu2GRieV9iVyxdY/w270-h320/kerucut9.png' alt=\"Pyramid-to-cone idea\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003ePyramid-to-cone idea\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw4JhLC8tVJJ9p8eLC_3XqK-yQBHv95Th6kmt7OblScK-UY6BYZsKwptTQsfwaIaXwCBc6V2wGEHEVKp4wVAJbDzncg9wERCEgJW9q7ISZq_iSSY2Vpz9bu04Bfrs7mu2GRieV9iVyxdY/s276/kerucut9.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw4JhLC8tVJJ9p8eLC_3XqK-yQBHv95Th6kmt7OblScK-UY6BYZsKwptTQsfwaIaXwCBc6V2wGEHEVKp4wVAJbDzncg9wERCEgJW9q7ISZq_iSSY2Vpz9bu04Bfrs7mu2GRieV9iVyxdY/s276/kerucut9.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eThe cone volume formula is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV=\\frac{1}{3}\\pi r^2t\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eUsing the diameter \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, it can also be written as:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV=\\frac{1}{12}\\pi d^2t\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-1-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h4\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er=9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003et=4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e339.12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV=\\frac{1}{3}\\times 3.14\\times 9^2\\times 4=339.12\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e339.12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the volume is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e339.12\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n339.12\\text{ cm}^3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e339.12\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-2-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h4\u003e\u003cp\u003eThe cone volume is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e314\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e314\\text{ cm}^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e314\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the base radius is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm. Find the slant height.\u003c/p\u003e\n\u003cp\u003eFirst find the height:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e314\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e⇒\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n314=\\frac{1}{3}\\times 3.14\\times 5^2\\times t\n\\Rightarrow t=12\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e314\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e⇒\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen find the slant height:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e169\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ns=\\sqrt{5^2+12^2}=\\sqrt{169}=13\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0223em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1777em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0839em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e169\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the slant height is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n13\\text{ cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example 3\"\u003e\u003c/a\u003eExample 3\u003c/h4\u003e\u003cp\u003eConsider the combined solid below.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj90DQhSJwOIRyllXgCLVeyP033sXYn8wLQOz90_2L2ZlpVmY6R_seOdJiUqTcR38KxEiYpjliR4jV3rF_SXjvvkL4EvlLgazEEY7njOmWafHEwJoRbCNOhzxcLHG76RUaa8pymbCvBoBc/w281-h320/kerucut7.png\" title=\"Cylinder and cone combination\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj90DQhSJwOIRyllXgCLVeyP033sXYn8wLQOz90_2L2ZlpVmY6R_seOdJiUqTcR38KxEiYpjliR4jV3rF_SXjvvkL4EvlLgazEEY7njOmWafHEwJoRbCNOhzxcLHG76RUaa8pymbCvBoBc/w281-h320/kerucut7.png\" data-sub-html=\"\u003ch2\u003eCylinder and cone combination\u003c/h2\u003e\u003cp\u003eCylinder and cone combination\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj90DQhSJwOIRyllXgCLVeyP033sXYn8wLQOz90_2L2ZlpVmY6R_seOdJiUqTcR38KxEiYpjliR4jV3rF_SXjvvkL4EvlLgazEEY7njOmWafHEwJoRbCNOhzxcLHG76RUaa8pymbCvBoBc/w281-h320/kerucut7.png' alt=\"Cylinder and cone combination\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eCylinder and cone combination\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj90DQhSJwOIRyllXgCLVeyP033sXYn8wLQOz90_2L2ZlpVmY6R_seOdJiUqTcR38KxEiYpjliR4jV3rF_SXjvvkL4EvlLgazEEY7njOmWafHEwJoRbCNOhzxcLHG76RUaa8pymbCvBoBc/s323/kerucut7.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj90DQhSJwOIRyllXgCLVeyP033sXYn8wLQOz90_2L2ZlpVmY6R_seOdJiUqTcR38KxEiYpjliR4jV3rF_SXjvvkL4EvlLgazEEY7njOmWafHEwJoRbCNOhzxcLHG76RUaa8pymbCvBoBc/s323/kerucut7.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eSuppose the cylinder volume is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003ecylinder\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1540\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV_{\\text{cylinder}}=\\pi r^2 t=\\frac{22}{7}\\times 7\\times 7\\times 10=1540\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3361em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003ecylinder\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1540\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAnd the cone volume is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003econe\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e462\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV_{\\text{cone}}=\\frac{1}{3}\\pi r^2 t=\\frac{1}{3}\\times \\frac{22}{7}\\times 7\\times 7\\times 9=462\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003econe\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e462\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTotal volume:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1540\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e462\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2002\u003c/mn\u003e\u003cmsup\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n1540+462=2002\\text{ cm}^3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1540\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e462\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2002\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"6-volume-comparison-when-the-radius-changes\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-volume-comparison-when-the-radius-changes\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Volume Comparison When the Radius Changes\"\u003e\u003c/a\u003e6. Volume Comparison When the Radius Changes\u003c/h3\u003e\u003cp\u003eIf the radius changes from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er+a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e while the height stays the same, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV:V_{\\text{new}}=r^2:(r+a)^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf the radius becomes \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e times the original, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV:V_{\\text{new}}=1:k^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eA cone has radius 2 cm.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eIf the radius increases by 1 cm:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV:V_{\\text{new}}=2^2:3^2=4:9\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eIf the radius becomes 3 times the original:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmtext\u003enew\u003c/mtext\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV:V_{\\text{new}}=1:3^2=1:9\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003enew\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"d-geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: D. GeoGebra Simulation\"\u003e\u003c/a\u003eD. GeoGebra Simulation\u003c/h2\u003e\u003cp\u003eUse the simulation below to observe the relationship between radius, height, slant height, surface area, and volume.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/ewqkcdjz/width/698/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false\" style=\"border: 0px;\" title=\"Luas Sisi dan Volume Kerucut\" width=\"650px\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"e-practice\" class=\"headerLink\"\u003e\n \u003ca href=\"#e-practice\" class=\"header-mark\" aria-label=\"Direct link to section: E. Practice\"\u003e\u003c/a\u003eE. Practice\u003c/h2\u003e\u003cp\u003eTry these questions:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eFind the surface area of a cone if the radius and height are known.\u003c/li\u003e\n\u003cli\u003eFind the volume of a cone if the diameter and height are known.\u003c/li\u003e\n\u003cli\u003eFind the slant height if the radius and volume are known.\u003c/li\u003e\n\u003cli\u003eFind the ratio of volumes when the radius changes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"f-learning-video\" class=\"headerLink\"\u003e\n \u003ca href=\"#f-learning-video\" class=\"header-mark\" aria-label=\"Direct link to section: F. Learning Video\"\u003e\u003c/a\u003eF. Learning Video\u003c/h2\u003e\u003cp\u003eYou can watch the supporting lesson video below:\u003c/p\u003e\n\u003ciframe allowfullscreen=\"\" height=\"315\" src=\"https://www.youtube.com/embed/zZ2pkHLTRJY\" title=\"Cone lesson video\" width=\"560\"\u003e\u003c/iframe\u003e\n", "language": "en" }, { "title" : "The Miracle of the Beehive", "date_published" : "2021-01-13T15:03:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/keajaiban-sarang-lebah/", "url" : "https://www.aantriono.com/en/keajaiban-sarang-lebah/", "summary": " The Architectural Genius of Honey BeesDear readers, did you know that honey bees are not only famous as small animals that produce honey, but are also famous as \"genius\" living creatures? As animals, honey bees live in colonies (group) in nests that they build themselves with great care. In each hive, there are thousands of hexagonal or hexagon-shaped bags made to store honey. Have we ever thought, why do they make them hexagonal shaped pockets.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"the-architectural-genius-of-honey-bees\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-architectural-genius-of-honey-bees\" class=\"header-mark\" aria-label=\"Direct link to section: The Architectural Genius of Honey Bees\"\u003e\u003c/a\u003eThe Architectural Genius of Honey Bees\u003c/h2\u003e\u003cp\u003eDear readers, did you know that honey bees are not only famous as small animals that produce honey, but are also famous as \"genius\" living creatures? As animals, honey bees live in colonies (group) in nests that they build themselves with great care. In each hive, there are thousands of hexagonal or hexagon-shaped bags made to store honey. Have we ever thought, why do they make them hexagonal shaped pockets.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr2C5Jf8bNf10grrMCWB4CKGlNYvoc-MxkxMoaPjH4Llm-_cVS2IDKFmipwex7G2fX8mXYlJMyGZ3vJ9RVv3eyQDWnnroFIeJM90Etq7Zqr7XNZrGKt7XIWxWNigB1kiUy_wK6-3W0SMk/s500/tawon.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"375\" data-original-width=\"500\" height=\"240\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr2C5Jf8bNf10grrMCWB4CKGlNYvoc-MxkxMoaPjH4Llm-_cVS2IDKFmipwex7G2fX8mXYlJMyGZ3vJ9RVv3eyQDWnnroFIeJM90Etq7Zqr7XNZrGKt7XIWxWNigB1kiUy_wK6-3W0SMk/w320-h240/tawon.jpg\" title=\"gambar sarang lebah\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"why-are-beehives-hexagonal\" class=\"headerLink\"\u003e\n \u003ca href=\"#why-are-beehives-hexagonal\" class=\"header-mark\" aria-label=\"Direct link to section: Why are Beehives Hexagonal?\"\u003e\u003c/a\u003eWhy are Beehives Hexagonal?\u003c/h2\u003e\u003cp\u003eMathematicians are also looking for answers to this question. Apparently, after doing long calculations, they found a very interesting answer. Namely, that the best way to build a storage warehouse with the largest capacity and using as few building materials as possible is to make hexagonal walls. Because according to them, if the bees build storage sacs in the form of tubes or triangular prisms, empty gaps will form between one sac and another. Moreover, less honey is stored in it. Triangular or square honey bags can be made without leaving any gaps, but mathematicians are aware of one important thing. Of all these geometric shapes, the one with the smallest perimeter is the hexagonal. For this reason, even though the shapes cover the same area, less material is required to construct a hexagonal compared to a square or triangle. This hexagonal cell/chamber design is very practical in many ways. The cells fit perfectly when arranged and share one wall together. Again, this ensures maximum storage with minimal materials. Apart from the side walls of the cells, bees also use this principle of maximum savings when building the bottom edges. The nest is made like a flat piece with two rows of cells back to back. In this case, a problem occurs at the meeting point of two cells. This problem can be solved by building the bottom surface of the cell by combining three squares. When three cells are built on one side of the nest, the bottom surface of the cells on the other side is automatically formed. Because the bottom surface is composed of square wax plates, the bottom of cells made in this way is deeper. This means the cell volume increases, and also means the amount of honey that can be stored increases. Then another question arises, why doesn\u0026rsquo;t honey leak or come out of the cell mouth? How do they keep the honey from coming out of the cell\u0026rsquo;s mouth?\u003c/p\u003e\n\u003ch2 id=\"the-significance-of-bees-in-the-quran\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-significance-of-bees-in-the-quran\" class=\"header-mark\" aria-label=\"Direct link to section: The Significance of Bees in the Qur\u0026rsquo;an\"\u003e\u003c/a\u003eThe Significance of Bees in the Qur\u0026rsquo;an\u003c/h2\u003e\u003cp\u003eAs we know, the food source for bees is nectar, which is not found in winter. Therefore, bees mix the nectar they collect in summer with a special fluid secreted by their bodies. This mixture produces a new nutritious substance, namely honey, and stores it for the coming winter. It is interesting to note that bees store much more honey than they actually need. The first question that comes to our mind is: why don\u0026rsquo;t the bees stop this excess production of honey. The answer is in the Qur\u0026rsquo;an Surah An Nahl verses 68-69. \u003cspan lang=\"ar\"\u003eالْجِبَالِ بُيُوتًا وَمِنَ الشَّجَرِ وَمِمَّا يَعْرِشُونَ. ثُمَّ كُلِي مِنْ كُلِِ لِقَوْمٍ يَتَفَكَّرُونَ\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eMeaning: And your Lord revealed to the bees: \u0026ldquo;Make nests in the hills, in the trees, and in places made by humans.\u0026rdquo; then eat from each (type) fruit and walk the path of your Lord which has been made easy (for you). Drink comes out of the bee\u0026rsquo;s stomach (honey) of various colors, in which there is medicine that heals humans. Verily, in that there is indeed a sign (greatness of God) for people who think. Indeed, Allah is great with all His creation, Allah is true with all His words. I hope this is helpful.🙏.\u003c/p\u003e", "language": "en" }, { "title" : "Earth Moon And Our Eyes", "date_published" : "2021-01-12T14:52:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bumi-bulan-dan-mata-kita/", "url" : "https://www.aantriono.com/en/bumi-bulan-dan-mata-kita/", "summary": " Astronomical and Anatomical Calculations Calculating the Earth\u0026rsquo;s MassEarth Earth is where we stand and is home to living creatures. Of course, living creatures that we know such as humans, animals, plants, fungi, microorganisms and so on. It is from this earth that humans try to study the celestial objects they see. So finally we know that Earth is one of the planets that surrounds the solar system.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"astronomical-and-anatomical-calculations\" class=\"headerLink\"\u003e\n \u003ca href=\"#astronomical-and-anatomical-calculations\" class=\"header-mark\" aria-label=\"Direct link to section: Astronomical and Anatomical Calculations\"\u003e\u003c/a\u003eAstronomical and Anatomical Calculations\u003c/h2\u003e\u003ch2 id=\"calculating-the-earths-mass\" class=\"headerLink\"\u003e\n \u003ca href=\"#calculating-the-earths-mass\" class=\"header-mark\" aria-label=\"Direct link to section: Calculating the Earth\u0026rsquo;s Mass\"\u003e\u003c/a\u003eCalculating the Earth\u0026rsquo;s Mass\u003c/h2\u003e\u003cp\u003e\u003cstrong\u003eEarth\u003c/strong\u003e Earth is where we stand and is home to living creatures. Of course, living creatures that we know such as humans, animals, plants, fungi, microorganisms and so on. It is from this earth that humans try to study the celestial objects they see. So finally we know that Earth is one of the planets that surrounds the solar system.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003cp\u003eEarth is the third planet from the sun, as well as the densest and fifth largest planet of the eight planets that revolve around the sun in the solar system.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003c!--more--\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlufEmDsPWzUSpRxehFbzWwwLgrnMz0vdUnCtxNgWSPMErk2TYSDm8Y1140zX1EBBX3phM8VbE84_aK1VgCqWuedJg5nIaDSyhlUvW45Xew2VikNWTpMyJr7t8eRH1FhfDQzYmNgM7g1M/s600/bumi-removebg-preview.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"400\" data-original-width=\"600\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlufEmDsPWzUSpRxehFbzWwwLgrnMz0vdUnCtxNgWSPMErk2TYSDm8Y1140zX1EBBX3phM8VbE84_aK1VgCqWuedJg5nIaDSyhlUvW45Xew2VikNWTpMyJr7t8eRH1FhfDQzYmNgM7g1M/s16000/bumi-removebg-preview.png\" title=\"gambar bumi\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThe Earth is considered to be a solid sphere with a mean radius of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6.37\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6.37\\times 10^6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.37\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m. The mass of the earth is expressed by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003eg\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eM=\\frac{r^2g}{G}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.415em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.07em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4461em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8913em;\"\u003e\u003cspan style=\"top:-2.931em;margin-right:0.0714em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size3 size1 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\"\u003eg\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e represents the radius of the earth, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eg\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eg\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eg\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the gravitational acceleration at the earth\u0026rsquo;s surface, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eG\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the general constant of gravity. The mass of the earth if \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eg\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9.8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eg=9.8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eg\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9.8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6.67\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eg\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eG=6.67\\times 10^{-11}NM^2kg^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.67\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eN\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10903em;\"\u003eM\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eg\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is:\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003eg\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6.37\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e9.8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e6.67\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6.37\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e9.8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e6.67\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6.37\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e9.8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e6.67\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e397\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e65362\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e6.67\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmn\u003e59\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e618\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e23\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e96\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{aligned}\nm\u0026amp;=\\frac{r^2g}{G}\\\\\n\u0026amp;=\\frac{(6.37\\times 10^6)^2 (9.8)}{6.67\\times 10^{-11}}\\\\\n\u0026amp;=\\frac{(6.37^2\\times 10^{12}) (9.8)}{6.67\\times 10^{-11}}\\\\\n\u0026amp;=\\frac{(6.37^2\\times 9.8)\\times 10^{12}}{6.67\\times 10^{-11}}\\\\\n\u0026amp;=\\frac{397, 65362\\times 10^{12}}{6.67\\times 10^{-11}}\\\\\n\u0026amp;\\approx 59, 618\\times 10^{23}\\\\\n\u0026amp;\\approx 5, 96\\times 10^{24}\n\\end{aligned}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:15.7671em;vertical-align:-7.6335em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:8.1335em;\"\u003e\u003cspan style=\"top:-10.1335em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-7.6564em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.096em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.5356em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.0249em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.9583em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:3.4824em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:7.6335em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:8.1335em;\"\u003e\u003cspan style=\"top:-10.1335em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eg\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-7.6564em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6.67\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.37\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e9.8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.096em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6.67\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e9.8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.5356em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6.67\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6.3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9.8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.0249em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6.67\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e397\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e65362\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.9583em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e59\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e618\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e23\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:3.4824em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e96\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:7.6335em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eSo, the mass of the earth is about \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5.96\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5.96\\times 10^{24}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5.96\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e kg.## Calculating the Moon\u0026rsquo;s Orbital Speed\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMonth\u003c/strong\u003e The moon is the earth\u0026rsquo;s satellite. The moon always rotates around the earth and around the sun. \u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdZg4FzzLoLYgs2m1IlRov7ECPQe4RUFr0f7JyXxfA0OpMiif08JMz84pCMtZO3rc0dsxwDw7WjtBELdcDRI_ol2hjPc1pmQd9k8sIp22RSvO_N1m8kK9RmVuMS7Lz8xT7gOWhxGxSBEs/s667/3107286900-removebg-preview.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"374\" data-original-width=\"667\" height=\"179\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdZg4FzzLoLYgs2m1IlRov7ECPQe4RUFr0f7JyXxfA0OpMiif08JMz84pCMtZO3rc0dsxwDw7WjtBELdcDRI_ol2hjPc1pmQd9k8sIp22RSvO_N1m8kK9RmVuMS7Lz8xT7gOWhxGxSBEs/w320-h179/3107286900-removebg-preview.png\" title=\"gambar bulan\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThe orbital speed of the moon is expressed by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ev=\\frac{2\\pi r}{T}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ev\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\"\u003eT\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e represents the average radius and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eT\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eT\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e represents the time (in seconds) required in one round. The orbital speed of the moon around the Earth with an average radius of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3.84\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3.84\\times 10^8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.84\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m and taking \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2.36\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2.36\\times 10^6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2.36\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e seconds to complete one revolution \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(\\pi=3.14)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is:\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3.84\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2.36\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e6.28\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3.84\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2.36\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e24.1152\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2.36\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e218\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmn\u003e1.02\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{aligned}\nv\u0026amp;=\\frac{2\\pi r}{T}\\\\\n\u0026amp;=\\frac{2(3.14) (3.84\\times 10^8)}{2.36\\times 10^6}\\\\\n\u0026amp;=\\frac{6.28 (3.84\\times 10^8)}{2.36\\times 10^6}\\\\\n\u0026amp;=\\frac{24.1152\\times 10^8}{2.36\\times 10^6}\\\\\n\u0026amp;\\approx 10,218\\times 10^2\\\\\n\u0026amp;\\approx 1.02\\times 10^3\n\\end{aligned}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:13.037em;vertical-align:-6.2685em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.7685em;\"\u003e\u003cspan style=\"top:-8.9382em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ev\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.461em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.9006em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.3402em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.5933em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:2.1174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.2685em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.7685em;\"\u003e\u003cspan style=\"top:-8.9382em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eT\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.461em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2.36\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.84\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.9006em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2.36\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6.28\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.84\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.3402em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2.36\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e24.1152\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.5933em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e218\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:2.1174em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1.02\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.2685em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eSo, the orbital speed of the moon around the Earth is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1.02\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1.02\\times 10^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1.02\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m/s.## Calculating the Pupil\u0026rsquo;s Area\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEye Pupils\u003c/strong\u003e The pupil is in the center of the iris (the part that gives color to the eyes). The function of the pupil is to allow light to enter the eye so that it can be focused on the retina to begin the vision process. In general, the pupils will look perfectly round, the same size and black. The black color occurs because light passing through the pupil is absorbed by the retina and is not reflected back.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEdvOXA5gMssK2bFKWDh08sa8kSx-m7rNNkD3JW9jyJkPPdyNuKr5ny18hTus1jBau5_J0CBGL5w32FadGKwkOpWBxl3heldv1uC2DVZl6eD1_u67XwIu31XB6zR73cAUi21MR2u-7uxo/s425/pupil.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"282\" data-original-width=\"425\" height=\"212\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEdvOXA5gMssK2bFKWDh08sa8kSx-m7rNNkD3JW9jyJkPPdyNuKr5ny18hTus1jBau5_J0CBGL5w32FadGKwkOpWBxl3heldv1uC2DVZl6eD1_u67XwIu31XB6zR73cAUi21MR2u-7uxo/w320-h212/pupil.jpg\" title=\"gambar pupil mata\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eWhen the light is dim, the pupil radius becomes \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\\times 10^{-3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m. Using the circle area formula \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eL=\\pi r^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the area of ​​the pupil in dim light with a value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi = 3.14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is:\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eL\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e113.04\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e130\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{aligned}\nL\u0026amp;=\\pi r^2\\\\\n\u0026amp;=3.14\\times (6\\times 10^{-3})^2\\\\\n\u0026amp;=3.14\\times 6^2\\times 10^{-6}\\\\\n\u0026amp;=113.04\\times 10^{-6}\\\\\n\u0026amp;=1,130\\times 10^{-4}\\\\\n\\end{aligned}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.6205em;vertical-align:-3.5603em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0603em;\"\u003e\u003cspan style=\"top:-6.1962em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.0997em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5603em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0603em;\"\u003e\u003cspan style=\"top:-6.1962em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e113.04\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.0997em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e130\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5603em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eSo, the area of ​​the pupil in dim light is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e130\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e1,130\\times 10^{-4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e130\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m. So, I hope this article is useful. If you have criticism, suggestions or improvements, please leave them in the comments. Thank you.😊🙏\u003c/p\u003e", "language": "en" }, { "title" : "Pythagorean Triples", "date_published" : "2021-01-11T14:26:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/tripel-pythagoras/", "url" : "https://www.aantriono.com/en/tripel-pythagoras/", "summary": " IntroductionFrom the Pythagorean theorem, we know the relation:\na2+b2=c2 a^2+b^2=c^2 a2+b2=c2Three positive integers (a,b,c)(a,b,c)(a,b,c) that satisfy this equation are called Pythagorean triples. If the three numbers have no common factor other than 1, the triple is called a primitive Pythagorean triple.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eFrom the \u003ca href=\"/2021/01/teorema-pythagoras/\" rel=\"\"\u003ePythagorean theorem\u003c/a\u003e, we know the relation:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^2+b^2=c^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThree positive integers \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a,b,c)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e that satisfy this equation are called \u003cstrong\u003ePythagorean triples\u003c/strong\u003e. If the three numbers have no common factor other than \u003ccode\u003e1\u003c/code\u003e, the triple is called a \u003cstrong\u003eprimitive Pythagorean triple\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"basic-formula\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-formula\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Formula\"\u003e\u003c/a\u003eBasic Formula\u003c/h2\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are positive integers with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u0026gt;n\u0026gt;0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then one standard form of a Pythagorean triple is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(a,b,c)=(2mn,\\ m^2-n^2,\\ m^2+n^2)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0585em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAB3SqXnUDjJ2I4Vo8MusXrxhqPnmMFB1JqW70ooMXBo2DnLepmdULYg-331Fe4M9bHkCgICH-FNHSgi16wRgoVXoeZLvAjYC249N-Ql6E_R4XWZ8pdHUKd7Oy1joyc28B0wdHVKZsqOfbvT_KEJNGfhf9Vgns8JhQL3S5NEDk8uN6khs6rKbSHNtt/s1280/triple.png' alt=\"Generating Pythagorean triples\" \u003e\u003c/p\u003e\n\u003ch2 id=\"proof\" class=\"headerLink\"\u003e\n \u003ca href=\"#proof\" class=\"header-mark\" aria-label=\"Direct link to section: Proof\"\u003e\u003c/a\u003eProof\u003c/h2\u003e\u003cp\u003eLet:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na \u0026amp;= 2mn\\\\\nb \u0026amp;= m^2-n^2\\\\\nc \u0026amp;= m^2+n^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na^2 \u0026amp;= 4m^2n^2\\\\\nb^2 \u0026amp;= (m^2-n^2)^2 = m^4+n^4-2m^2n^2\\\\\na^2+b^2 \u0026amp;= 4m^2n^2 + m^4+n^4-2m^2n^2\\\\\n\u0026amp;= m^4+2m^2n^2+n^4\\\\\n\u0026amp;= (m^2+n^2)^2\\\\\n\u0026amp;= c^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:9.1446em;vertical-align:-4.3223em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.8223em;\"\u003e\u003cspan style=\"top:-6.9582em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.4341em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan 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style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8618em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.6623em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.3223em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.8223em;\"\u003e\u003cspan style=\"top:-6.9582em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.4341em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8618em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.6623em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.3223em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2mn,\\ m^2-n^2,\\ m^2+n^2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e really is a Pythagorean triple.\u003c/p\u003e\n\u003ch2 id=\"area-and-perimeter\" class=\"headerLink\"\u003e\n \u003ca href=\"#area-and-perimeter\" class=\"header-mark\" aria-label=\"Direct link to section: Area and Perimeter\"\u003e\u003c/a\u003eArea and Perimeter\u003c/h2\u003e\u003cp\u003eIf the legs and hypotenuse of a right triangle are written in this form, then:\u003c/p\u003e\n\u003ch3 id=\"area\" class=\"headerLink\"\u003e\n \u003ca href=\"#area\" class=\"header-mark\" aria-label=\"Direct link to section: Area\"\u003e\u003c/a\u003eArea\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{1}{2}(2mn)(m^2-n^2)=mn(m^2-n^2)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"perimeter\" class=\"headerLink\"\u003e\n \u003ca href=\"#perimeter\" class=\"header-mark\" aria-label=\"Direct link to section: Perimeter\"\u003e\u003c/a\u003ePerimeter\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2mn+(m^2-n^2)+(m^2+n^2)=2mn+2m^2=2m(m+n)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIn some cases, the area and the perimeter are equal. Starting from:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nmn(m^2-n^2)=2m(m+n)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewe can derive:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nn(m-n)=2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eExamples:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eif \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then area = perimeter = \u003ccode\u003e24\u003c/code\u003e\u003c/li\u003e\n\u003cli\u003eif \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then area = perimeter = \u003ccode\u003e30\u003c/code\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"types-of-pythagorean-triples\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-pythagorean-triples\" class=\"header-mark\" aria-label=\"Direct link to section: Types of Pythagorean Triples\"\u003e\u003c/a\u003eTypes of Pythagorean Triples\u003c/h2\u003e\u003ch3 id=\"1-primitive-pythagorean-triples\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-primitive-pythagorean-triples\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Primitive Pythagorean Triples\"\u003e\u003c/a\u003e1. Primitive Pythagorean Triples\u003c/h3\u003e\u003cp\u003eA Pythagorean triple \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a,b,c)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called \u003cstrong\u003eprimitive\u003c/strong\u003e if:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003egcd\u003c/mi\u003e\u003cmo\u003e⁡\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\gcd(a,b,c)=1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mop\"\u003e\u003cspan style=\"margin-right:0.01389em;\"\u003eg\u003c/span\u003ecd\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSome useful properties:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eat least one of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is odd\u003c/li\u003e\n\u003cli\u003eone of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is even\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is always odd\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"2-twin-primitive-pythagorean-triples\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-twin-primitive-pythagorean-triples\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Twin Primitive Pythagorean Triples\"\u003e\u003c/a\u003e2. Twin Primitive Pythagorean Triples\u003c/h3\u003e\u003cp\u003eA primitive triple is called \u003cstrong\u003etwin primitive\u003c/strong\u003e when the difference between the hypotenuse and the longer leg is \u003ccode\u003e1\u003c/code\u003e.\u003c/p\u003e\n\u003cp\u003eUsing:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na \u0026amp;= 2mn\\\\\nb \u0026amp;= m^2-n^2\\\\\nc \u0026amp;= m^2+n^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eif \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u0026gt;b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the twin condition gives:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nc \u0026amp;= a+1\\\\\nm^2+n^2 \u0026amp;= 2mn+1\\\\\n(m-n)^2 \u0026amp;= 1\\\\\nm-n \u0026amp;= 1\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0482em;vertical-align:-2.7741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.4341em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" 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class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.6841em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhich has no positive integer solution.\u003c/p\u003e\n\u003ch3 id=\"3-composite-pythagorean-triples\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-composite-pythagorean-triples\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Composite Pythagorean Triples\"\u003e\u003c/a\u003e3. Composite Pythagorean Triples\u003c/h3\u003e\u003cp\u003eA triple is called \u003cstrong\u003ecomposite\u003c/strong\u003e when the three numbers still have a common factor greater than \u003ccode\u003e1\u003c/code\u003e.\u003c/p\u003e\n\u003ch2 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h2\u003e\u003cp\u003eProve that if one of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is even in a primitive Pythagorean triple \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a,b,c)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e must be odd.\u003c/p\u003e\n\u003ch3 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eUse an indirect proof. Suppose \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is even while \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are odd.\u003c/p\u003e\n\u003cp\u003eWrite:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na \u0026amp;= 2p+1\\\\\nb \u0026amp;= 2m+1\\\\\nc \u0026amp;= 2n\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSubstitute into the Pythagorean equation:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na^2+b^2 \u0026amp;= c^2\\\\\n(2p+1)^2 + (2m+1)^2 \u0026amp;= (2n)^2\\\\\n4p^2+4p+1 + 4m^2+4m+1 \u0026amp;= 4n^2\\\\\np^2+p+m^2+m+\\frac{1}{2} \u0026amp;= n^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.8798em;vertical-align:-3.1899em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6899em;\"\u003e\u003cspan style=\"top:-6.1472em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6231em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.099em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.1176em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1899em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6899em;\"\u003e\u003cspan style=\"top:-6.1472em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6231em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.099em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.1176em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1899em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe left-hand side is not an integer, but \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e must be an integer. This contradiction shows that \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cannot be even. Therefore, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e must be odd.\u003c/p\u003e\n\u003ch2 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h2\u003e\u003cp\u003eFind all primitive Pythagorean triples \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(a,b,c)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e even and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec \\le 40\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7719em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"solution-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-1\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eTake:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na \u0026amp;= m^2-n^2\\\\\nb \u0026amp;= 2mn\\\\\nc \u0026amp;= m^2+n^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith conditions:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u0026gt;n\u0026gt;0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are relatively prime\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are not both odd\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e≤\u003c/mo\u003e\u003cmn\u003e40\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em^2+n^2 \\le 40\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9501em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≤\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe valid results are:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=m^2-n^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=2mn\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=m^2+n^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003eTriple\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(3,4,5)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e12\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e13\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(5,12,13)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e15\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e8\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e17\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e17\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(15,8,17)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e17\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e7\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e24\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e25\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(7,24,25)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e21\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e20\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e29\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e29\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(21,20,29)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e29\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e6\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e35\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e12\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e37\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e35\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e37\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(35,12,37)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e35\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e37\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"how-to-generate-pythagorean-triples\" class=\"headerLink\"\u003e\n \u003ca href=\"#how-to-generate-pythagorean-triples\" class=\"header-mark\" aria-label=\"Direct link to section: How to Generate Pythagorean Triples\"\u003e\u003c/a\u003eHow to Generate Pythagorean Triples\u003c/h2\u003e\u003cp\u003eGeneral steps:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003echoose two natural numbers \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u0026gt;n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003ecompute \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em^2-n^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2mn\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em^2+n^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003ethose three values form a Pythagorean triple\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em^2-n^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2mn\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003emn\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n \u003cth style=\"text-align: right\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em^2+n^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/th\u003e\n 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encoding=\"application/x-tex\"\u003e(3,4,5)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e1\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e8\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e6\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e10\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(6,8,10)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e2\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e12\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e13\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(5,12,13)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e3\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e7\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e24\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e25\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(7,24,25)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e4\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e9\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e40\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e41\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e40\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e41\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(9,40,41)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e40\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e41\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: right\"\u003e6\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e11\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e60\u003c/td\u003e\n \u003ctd style=\"text-align: right\"\u003e61\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e61\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(11,60,61)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e61\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eNotes:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eevery multiple of a Pythagorean triple is also a Pythagorean triple\u003c/li\u003e\n\u003cli\u003ePythagorean triples are not limited to integers; they also appear with rational or irrational numbers, such as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(3,3,3\\sqrt{2})\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1572em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 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separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\left(\\frac{3}{2},2,\\frac{5}{2}\\right)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003ePythagorean triples provide a fast way to build right triangles without starting from scratch every time. Once the basic formula and the primitive conditions are understood, many useful numerical patterns become easy to generate.\u003c/p\u003e\n", "language": "en" }, { "title" : "Pythagorean theorem", "date_published" : "2021-01-10T15:23:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/teorema-pythagoras/", "url" : "https://www.aantriono.com/en/teorema-pythagoras/", "summary": " IntroductionIn a right triangle, there is a special relationship between the hypotenuse and the two legs. This relationship is known as the Pythagorean theorem.\nBasic Competencies3.6 Explain and prove the Pythagorean theorem and Pythagorean triples.\n4.6 Solve problems related to the Pythagorean theorem and Pythagorean triples.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eIn a right triangle, there is a special relationship between the hypotenuse and the two legs. This relationship is known as the Pythagorean theorem.\u003c/p\u003e\n\u003ch2 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h2\u003e\u003cp\u003e3.6 Explain and prove the Pythagorean theorem and Pythagorean triples.\u003c/p\u003e\n\u003cp\u003e4.6 Solve problems related to the Pythagorean theorem and Pythagorean triples.\u003c/p\u003e\n\u003ch2 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying this material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eVerify the Pythagorean theorem.\u003c/li\u003e\n\u003cli\u003eDetermine an unknown side of a right triangle when two sides are known.\u003c/li\u003e\n\u003cli\u003eDetermine the type of a triangle from its side lengths.\u003c/li\u003e\n\u003cli\u003eTest whether three numbers form a Pythagorean triple.\u003c/li\u003e\n\u003cli\u003eApply the theorem to real-life problems.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h2\u003e\u003ch3 id=\"1-discovering-the-pythagorean-theorem\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-discovering-the-pythagorean-theorem\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Discovering the Pythagorean Theorem\"\u003e\u003c/a\u003e1. Discovering the Pythagorean Theorem\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWfRmlY_tWOzk4jNwJo346UOnsHpg23P-pz67Kk1AbnWocbDYRUTbx88Bpp6t9m8AGfCgm1AZ-F__uSv08WRAYSn_nzOfryUU-sIQo2nHWO7teqcw_xjGL3TQNEjloH1SwWVjOwzy-zw__oE4DpzTLyP93Wn4BIzbWNLuIcpvwJbotXT8UKDpsny1Q/w640-h200/pytha.png' alt=\"Pythagorean theorem\" \u003e\u003c/p\u003e\n\u003cp\u003eTriangle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABC\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is right-angled at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. The side opposite the right angle is called the \u003cstrong\u003ehypotenuse\u003c/strong\u003e, while the other two sides are called the legs.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrP-Qkx1WZH0xslQltoLEZMv0XFNwmhTmQDL2GINpXBrGf20Ng7J28AB_iBxuWnib-it7Xx8V1a4iXHn4kLDeLWv9HSiXc0b9PmbvKH7kXIUKNvpFie39AJqJAiP4DArOeQO_9n0dZZQs/w400-h193/pythagoras3.png' alt=\"Proof idea\" \u003e\u003c/p\u003e\n\u003cp\u003eFrom the square arrangements in the figure:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003earea of the large square\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003earea of the large square\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{area of the large square} \u0026amp;= a^2 + 2bc \\\\\n\\text{area of the large square} \u0026amp;= b^2 + c^2 + 2bc\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0482em;vertical-align:-1.2741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003earea of the large square\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003earea of the large square\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince both large squares have the same area:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na^2 + 2bc \u0026amp;= b^2 + c^2 + 2bc \\\\\na^2 \u0026amp;= b^2 + c^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.0482em;vertical-align:-1.2741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.7741em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoqLI5NuuHENuFhhlnt0Nc_pxKjDdhj985kBX_Oe2_Zop4mIKAuuqqYgd4-3csynfV15ruzjh2HLk1sZTkO5Oblgg6I1XAfg18eHi5twOOURGpldoPSohQ3t-IijZVOtJilpJns0Jq0_Q/s320/pythagoras4.png' alt=\"Area interpretation 1\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY62ovE_pGp7q8nv5XdiUzQgisMt9pJ8LyZTT-guhHrxONick0GMVnyZTtNuZxizu07-mb0Fsb0u_bRBHB4kAzVTpk9B-ppWONOJU9-mlzr0XXWmTRPPSjBO30xmYDGNbIzAqoILovT7Y/s269/pythagoras5.png' alt=\"Area interpretation 2\" \u003e\u003c/p\u003e\n\u003cp\u003eThis means that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDYSa2S8exx4qzcN6tXvgHOp9nwKG5Z62R1-gEoUUzCeb1uNF7Zv97SM6S9Eah-dk6FnE6zZbJa-VcY1bLCaK1B2GBzkmvUGmQUgq0C8uNjGfuW2tXkd-YTMqAt5P5-sgVXZjbiZnckrY/w200-h152/pythagoras6.png' alt=\"Pythagorean formula\" \u003e\u003c/p\u003e\n\u003cp\u003eIf the hypotenuse is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the legs are \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^2 = b^2 + c^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na = \\sqrt{b^2 + c^2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0223em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 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display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nb = \\sqrt{a^2 - c^2}\n\\qquad\nc = \\sqrt{a^2 - b^2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0223em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 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class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.1777em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.0623em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3.2em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:1em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.0223em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.2em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1777em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-using-the-pythagorean-theorem\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-using-the-pythagorean-theorem\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Using the Pythagorean Theorem\"\u003e\u003c/a\u003e2. Using the Pythagorean Theorem\u003c/h3\u003e\u003cp\u003eThe theorem can be used to determine an unknown side of a right triangle when the other two sides are known.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgc3Lzi_ryIkPCPtBuUUntcS93CznkTFFc5-oEVLmsJtsgtD_nSXfMDu0SSOuuXfMNjChV1k2uW8jITXsNpP9dz7y__r8gw-XFm_niGitkPsfp0swqmzSJd95NQ06o2sUH3qwbzjBvJ8ro/w400-h171/pythagoras7.png' alt=\"Examples of side calculation\" \u003e\u003c/p\u003e\n\u003ch4 id=\"solutions\" class=\"headerLink\"\u003e\n \u003ca href=\"#solutions\" class=\"header-mark\" aria-label=\"Direct link to section: Solutions\"\u003e\u003c/a\u003eSolutions\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eFor figure (a):\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e169\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e144\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\na^2 \u0026amp;= 13^2 - 12^2 \\\\\n \u0026amp;= 169 - 144 \\\\\n \u0026amp;= 25 \\\\\na \u0026amp;= 5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0241em;vertical-align:-2.7621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e169\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e144\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003eFor figure (b):\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e17\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e289\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e225\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e64\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nb^2 \u0026amp;= 17^2 - 15^2 \\\\\n \u0026amp;= 289 - 225 \\\\\n \u0026amp;= 64 \\\\\nb \u0026amp;= 8\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0241em;vertical-align:-2.7621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e289\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e225\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e64\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"3\"\u003e\n\u003cli\u003eFor figure (c):\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e576\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e49\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e625\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nc^2 \u0026amp;= 24^2 + 7^2 \\\\\n \u0026amp;= 576 + 49 \\\\\n \u0026amp;= 625 \\\\\nc \u0026amp;= 25\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0241em;vertical-align:-2.7621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e576\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e49\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e625\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-determining-the-type-of-triangle-from-its-sides\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-determining-the-type-of-triangle-from-its-sides\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Determining the Type of Triangle from Its Sides\"\u003e\u003c/a\u003e3. Determining the Type of Triangle from Its Sides\u003c/h3\u003e\u003ch4 id=\"a-converse-of-the-pythagorean-theorem\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-converse-of-the-pythagorean-theorem\" class=\"header-mark\" aria-label=\"Direct link to section: a. Converse of the Pythagorean Theorem\"\u003e\u003c/a\u003ea. Converse of the Pythagorean Theorem\u003c/h4\u003e\u003cp\u003eIf a triangle satisfies:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^2 = b^2 + c^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethen the triangle is a right triangle.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicUP4lqcYkazt1RygXSbh7Uca9EOyx7qiG-Bv2DpDZvcjuIrjvjPmqZYgM9xppvEMv_hlfeY8ofDV5VAd9EPgzpYWPtXe3dlPfQXucsdXnq-nMOuYGxnuHJjikxZChYt0nnudXQzgl-YI/w320-h193/pythagoras8.png' alt=\"Converse of Pythagoras\" \u003e\u003c/p\u003e\n\u003cp\u003eIn general, if \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the longest side:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec^2 = a^2 + b^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the triangle is \u003cstrong\u003eright\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec^2 \u0026gt; a^2 + b^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8532em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the triangle is \u003cstrong\u003eobtuse\u003c/strong\u003e.\u003c/li\u003e\n\u003cli\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e\u0026lt;\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec^2 \u0026lt; a^2 + b^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8532em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026lt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the triangle is \u003cstrong\u003eacute\u003c/strong\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"examples\" class=\"headerLink\"\u003e\n \u003ca href=\"#examples\" class=\"header-mark\" aria-label=\"Direct link to section: Examples\"\u003e\u003c/a\u003eExamples\u003c/h4\u003e\u003col\u003e\n\u003cli\u003eA triangle has side lengths \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e12\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e13\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e13\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e169\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmsup\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e144\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e169\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n13^2 = 169,\\qquad 12^2 + 5^2 = 144 + 25 = 169\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0585em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e169\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e144\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e169\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo it is a right triangle.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eA triangle has side lengths \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e81\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e49\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e74\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n9^2 = 81,\\qquad 7^2 + 5^2 = 49 + 25 = 74\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0585em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e81\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e49\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e74\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eBecause \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e81\u003c/mn\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmn\u003e74\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e81 \u0026gt; 74\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6835em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e81\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e74\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the triangle is obtuse.\u003c/p\u003e\n\u003ch4 id=\"b-pythagorean-triples\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-pythagorean-triples\" class=\"header-mark\" aria-label=\"Direct link to section: b. Pythagorean Triples\"\u003e\u003c/a\u003eb. Pythagorean Triples\u003c/h4\u003e\u003cp\u003eA Pythagorean triple is a set of three natural numbers that can be the side lengths of a right triangle.\u003c/p\u003e\n\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e15\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e20\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e25\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e225\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e400\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e625\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n15^2 + 20^2 = 225 + 400 = 625 = 25^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e225\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e400\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e625\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo they form a Pythagorean triple.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e20\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e24\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e29\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e29\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e29\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e400\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e576\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e976\u003c/mn\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e29\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n20^2 + 24^2 = 400 + 576 = 976 \\neq 29^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e400\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e576\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e976\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo they do not form a Pythagorean triple.\u003c/p\u003e\n\u003ch3 id=\"4-side-ratios-in-special-right-triangles\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-side-ratios-in-special-right-triangles\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Side Ratios in Special Right Triangles\"\u003e\u003c/a\u003e4. Side Ratios in Special Right Triangles\u003c/h3\u003e\u003ch4 id=\"a-isosceles-right-triangle\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-isosceles-right-triangle\" class=\"header-mark\" aria-label=\"Direct link to section: a. Isosceles right triangle\"\u003e\u003c/a\u003ea. Isosceles right triangle\u003c/h4\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgDFUcio8BYcNsP7uzCgAqnxAXaWc2LFWx7OfrceqW1mARYEXij0EEHpuhyL40LEncSjjDT5ddkqeBBVFZOeeRN1e5ZGDeGkq8zbq6PY4f1UkJEQBreK5gN-Lfq5dinroBEbarCPojOxQ/w200-h198/pythagoras9.png' alt=\"Isosceles right triangle\" \u003e\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB = BC = a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" 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class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0839em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh91VVVrcmq9nPJcSl4YE9m3qvdcRLPzunXGTvoG5DRXfZxTHU1aBvqtK0w6ZIAsYvvlMbBz2L1Zc-5qG5n1Ck1R_vAODIL6zsJy4vRw8wqGaqRmL7lhWFp10GDSgFf_lnndDA09uJ0iFY/w200-h168/pythagoras10.png' alt=\"45 degree example\" \u003e\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAC = 8\\sqrt{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nAB = BC = 8\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"b-right-triangle-with-a--angle\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-right-triangle-with-a--angle\" class=\"header-mark\" aria-label=\"Direct link to section: b. Right triangle with a 30∘30^\\circ30∘ angle\"\u003e\u003c/a\u003eb. Right triangle with a \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e30^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e angle\u003c/h4\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiSGGXsdmmgkX4hAKsa-t7R53QbwKKaPDxuXNTC2GuK2UK7kJioKVKKuDRyEO8T-cBbrRvcGdIPvkq1NLHf3Maw9jI-CZnoGxLrUHbYYKEmQiZnEOsR2BBn5vfm8jtFsTxGfYspRGDMCA/w320-h285/pythagoras11.png' alt=\"30 degree triangle\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4D2GzJN08JA9LykvST80HTDeliZ-lNHo79OgxQZDCxqIq8DhvAyqSlenAuLeUSaPC1dqhyphenhyphen880V4VHCDiQcCHv3uCNTWxkZ9DiAQW67ptV_98RKvD8kSsj4BSxAiVtpKmqntrsYuzUnY4/w320-h241/pythagoras12.png' alt=\"60 degree triangle\" \u003e\u003c/p\u003e\n\u003cp\u003eFrom the construction:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e:\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nBC : AC : AB = 1 : 2 : \\sqrt{3}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e:\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.0839em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eMeaning:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe side opposite \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e∘\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e30^\\circ\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6741em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mbin mtight\"\u003e∘\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the shortest side,\u003c/li\u003e\n\u003cli\u003ethe hypotenuse is twice that side,\u003c/li\u003e\n\u003cli\u003ethe remaining leg is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\sqrt{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e times that side.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn2bUQziiNKv8MIUb0VNi-bAXw8bCci-k9YBc6RtAccpGXM2TdVzkzSvNDvaw78i2Kqglnrh6EdXQmGJiucgUncUt_aELuIy7mEc6YhyphenhyphennP76uZWDoLYdMOW1ITxlNSk-tN2pD0oaQPo6w/w320-h229/pythagoras13.png' alt=\"30 degree example\" \u003e\u003c/p\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eAB = 3\\sqrt{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nAC = 6,\\qquad BC = 3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"5-face-diagonals-and-space-diagonals-of-a-cube\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-face-diagonals-and-space-diagonals-of-a-cube\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Face Diagonals and Space Diagonals of a Cube\"\u003e\u003c/a\u003e5. Face Diagonals and Space Diagonals of a Cube\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2Vo_CBW_FYvkwB74ABQDnTuQ5wUvOKyiYwU1zgx_q1uuz4s9gHMprrqfO3gRX0McL7zZjzj2DDuoirH_2pmfF1RS0IYfGr2ilrUeVPNZIAqEJ0Pm8sxyjm0pAH7HrwIkDJ_hj2y7rbKA/w320-h301/pythagoras14.png' alt=\"Cube diagonals\" \u003e\u003c/p\u003e\n\u003cp\u003eIf the edge length of a cube is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then the face diagonal is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nBG^2 \u0026amp;= BC^2 + CG^2 \\\\\n \u0026amp;= a^2 + a^2 \\\\\nBG \u0026amp;= a\\sqrt{2}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.6643em;vertical-align:-2.0822em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5822em;\"\u003e\u003cspan style=\"top:-4.718em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1939em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5778em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord 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style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath 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class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0822em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe space diagonal is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation 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class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1939em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5778em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9561em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.9161em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.0839em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0822em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo for a cube with edge length \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eface diagonal = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsqrt\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\\sqrt{2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003espace diagonal = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsqrt\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\\sqrt{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.1328em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9072em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8672em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1328em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"6-solving-word-problems\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-solving-word-problems\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Solving Word Problems\"\u003e\u003c/a\u003e6. Solving Word Problems\u003c/h3\u003e\u003cp\u003eGeneral steps for word problems using the Pythagorean theorem:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eRead the problem carefully.\u003c/li\u003e\n\u003cli\u003eDraw a sketch.\u003c/li\u003e\n\u003cli\u003eIdentify the known and unknown sides.\u003c/li\u003e\n\u003cli\u003eUse the appropriate formula.\u003c/li\u003e\n\u003cli\u003eCheck the result.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h4\u003e\u003cp\u003eAndi leans a ladder of length \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m against a tree. The bottom of the ladder is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m from the base of the tree. How high is the top of the ladder above the ground?\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi59cQUP6pGBsIro0dTYX-uXwT5MjNt8mg6tcgKCRmv2NIzwliwc60_zpjFAqj0x0f3APBGdnUYHdsCSjFC9l1SGlXWH-Lb92KCfjhUYpwpsCGMC2oNtcfm2PeAJhwL8Wuchn4q5UxHX5Q/w320-h278/pythagoras15.png' alt=\"Ladder problem\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eh\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n5^2 \u0026amp;= 3^2 + h^2 \\\\\n25 \u0026amp;= 9 + h^2 \\\\\nh^2 \u0026amp;= 16 \\\\\nh \u0026amp;= 4\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0723em;vertical-align:-2.7862em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2862em;\"\u003e\u003cspan style=\"top:-5.4221em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3738em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8738em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7862em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2862em;\"\u003e\u003cspan style=\"top:-5.4221em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3738em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8738em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7862em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the height is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e m.\u003c/p\u003e\n\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h4\u003e\u003cp\u003eA ship sails west for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e km and then south for \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e km. How far is it now from its starting point?\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWd38kvJGh3LcALKniOuHSpWLsYwVTCKP2Ay2K_fDIV2pH_NrMJ_Q93nkRdCLdHIKTfmbbkpptL6In1rqtNj_ABdytG3vtF1PtXbCRzLO6MxSs9HMocoee1vZZl7UzhMY-SX1rRPYvIGA/w320-h286/pythagoras16.png' alt=\"Ship problem\" \u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e36\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e64\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx^2 \u0026amp;= 6^2 + 8^2 \\\\\n \u0026amp;= 36 + 64 \\\\\n \u0026amp;= 100 \\\\\nx \u0026amp;= 10\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0241em;vertical-align:-2.7621em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2621em;\"\u003e\u003cspan style=\"top:-5.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e64\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7621em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the ship is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e10\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e km from its starting point.\u003c/p\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eThe old interactive worksheet in this article depended on a third-party embed that is no longer stable with the current blog theme. It has been removed so the page stays intact and readable.\u003c/p\u003e\n\u003cp\u003eIt can be replaced later with:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ea stable Hugo shortcode,\u003c/li\u003e\n\u003cli\u003ean external worksheet link,\u003c/li\u003e\n\u003cli\u003eor a local quiz component.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThis completes the Pythagorean theorem lesson. Corrections and suggestions can be shared through the available feedback channel.\u003c/p\u003e\n", "language": "en" }, { "title" : "Curved-Surface Solids", "date_published" : "2021-01-05T04:42:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/2021/01/bangun-ruang-sisi-lengkung/", "url" : "https://www.aantriono.com/2021/01/bangun-ruang-sisi-lengkung/", "summary": " IntroductionCurved-surface solids include cylinders, cones, and spheres. This article focuses on the cylinder, a solid formed by two parallel congruent circles and one curved lateral surface.\nA. Basic Competencies3.7 Generalize the surface area and volume of curved-surface solids such as cylinders, cones, and spheres.\n4.7 Solve contextual problems related to the surface area and volume of curved-surface solids and their combinations.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eCurved-surface solids include cylinders, cones, and spheres. This article focuses on the \u003cstrong\u003ecylinder\u003c/strong\u003e, a solid formed by two parallel congruent circles and one curved lateral surface.\u003c/p\u003e\n\u003ch2 id=\"a-basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: A. Basic Competencies\"\u003e\u003c/a\u003eA. Basic Competencies\u003c/h2\u003e\u003cp\u003e\u003ccode\u003e3.7\u003c/code\u003e Generalize the surface area and volume of curved-surface solids such as cylinders, cones, and spheres.\u003cbr\u003e\n\u003ccode\u003e4.7\u003c/code\u003e Solve contextual problems related to the surface area and volume of curved-surface solids and their combinations.\u003c/p\u003e\n\u003ch2 id=\"b-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: B. Learning Objectives\"\u003e\u003c/a\u003eB. Learning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying this lesson, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eidentify the elements of a cylinder\u003c/li\u003e\n\u003cli\u003ecalculate the base area, lateral area, and top area\u003c/li\u003e\n\u003cli\u003ecalculate the volume of a cylinder\u003c/li\u003e\n\u003cli\u003edetermine cylinder elements when the volume is known\u003c/li\u003e\n\u003cli\u003ecompare cylinder volumes after a change in radius\u003c/li\u003e\n\u003cli\u003ecalculate the change in cylinder volume when the radius changes\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-lesson-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-lesson-material\" class=\"header-mark\" aria-label=\"Direct link to section: C. Lesson Material\"\u003e\u003c/a\u003eC. Lesson Material\u003c/h2\u003e\u003ch3 id=\"1-elements-of-a-cylinder\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-elements-of-a-cylinder\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Elements of a Cylinder\"\u003e\u003c/a\u003e1. Elements of a Cylinder\u003c/h3\u003e\u003cp\u003eA cylinder is a curved-surface solid formed by two identical parallel circles and a rectangle wrapped around them.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwh-SKF1kguiKXqU9eCsAWfOSfC698WZMkhfY8XwPRrBDYTKbLjlK9BgEPsQ1ey1KkPxjaSrPdIi_8sHFR_wxAnLbitx0oNJDNjO7zO9EmxI0O5sZkMCKi_PbtlA3alTcqsOZswWKH18cWUBBKeNzO6OyBkovRTXeRTuCJxYEBiCSnJ9-uqSZpGu0e/s1280/tabung.png' alt=\"Calculating the area and volume of a cylinder\" \u003e\u003c/p\u003e\n\u003cp\u003eFrom the picture we see:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ea cylinder has three surfaces: the base, the top, and the curved side\u003c/li\u003e\n\u003cli\u003ethe base and top are circles\u003c/li\u003e\n\u003cli\u003ethe curved side is called the \u003cstrong\u003elateral surface\u003c/strong\u003e\u003c/li\u003e\n\u003cli\u003ethe distance between base and top is the \u003cstrong\u003eheight\u003c/strong\u003e, denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003et\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003ethe radius is denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003er\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the diameter by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ed\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"2-cylinder-net\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-cylinder-net\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Cylinder Net\"\u003e\u003c/a\u003e2. Cylinder Net\u003c/h3\u003e\u003cp\u003eA cylinder net consists of:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003etwo congruent circles\u003c/li\u003e\n\u003cli\u003eone rectangle\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN4xvEahw4dwpHEaPjtzud0MbBz8iJA1354VNxNzzQOS372PaOJrxv_lmdh5x6sdxEBTdktvoqF1IiZIswSO1GW2CXiVxxUzNRxw86XgH7FU7u9xpvl4ldJWDOOjDF3WT_oGSCSiwOUXM/s748/jaring+tabung.png' alt=\"Cylinder net\" \u003e\u003c/p\u003e\n\u003cp\u003eThe net is useful because the surface area of the cylinder is equal to the total area of this unfolded figure.\u003c/p\u003e\n\u003ch3 id=\"3-surface-area-of-a-cylinder\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-surface-area-of-a-cylinder\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Surface Area of a Cylinder\"\u003e\u003c/a\u003e3. Surface Area of a Cylinder\u003c/h3\u003e\u003cp\u003eThe surface area is the sum of:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ethe area of the base circle\u003c/li\u003e\n\u003cli\u003ethe area of the lateral surface\u003c/li\u003e\n\u003cli\u003ethe area of the top circle\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eDetails:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ebase area = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi r^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003elateral area = circumference of base \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e×\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\times\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e×\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e height = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\pi rt\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003etop area = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi r^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nS \u0026amp;= \\pi r^2 + 2\\pi rt + \\pi r^2\\\\\n \u0026amp;= 2\\pi r^2 + 2\\pi rt\\\\\n \u0026amp;= 2\\pi r(r+t)\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5482em;vertical-align:-2.0241em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5241em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6359em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0241em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eA cylinder has height \u003ccode\u003e13 cm\u003c/code\u003e and base radius \u003ccode\u003e7 cm\u003c/code\u003e. Find its surface area.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e44\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e880\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nS \u0026amp;= 2\\pi r(r+t)\\\\\n \u0026amp;= 2 \\times \\frac{22}{7} \\times 7 \\times (7+13)\\\\\n \u0026amp;= 44 \\times 20\\\\\n \u0026amp;= 880\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.8074em;vertical-align:-3.1537em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6537em;\"\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.6537em;\"\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.1537em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e44\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8277em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e880\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.1537em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e880\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n880\\ \\text{cm}^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e880\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"4-volume-of-a-cylinder\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-volume-of-a-cylinder\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Volume of a Cylinder\"\u003e\u003c/a\u003e4. Volume of a Cylinder\u003c/h3\u003e\u003cp\u003eSince the base is a circle, the volume of a cylinder is the base area times the height.\u003c/p\u003e\n\u003cp\u003eFormula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV = \\pi r^2 t\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor, in terms of diameter:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV = \\frac{1}{4}\\pi d^2 t\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h4\u003e\u003cp\u003eA cylinder has radius \u003ccode\u003e14 cm\u003c/code\u003e and height \u003ccode\u003e20 cm\u003c/code\u003e. Find its volume.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eV\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e14\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12,320\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nV \u0026amp;= \\pi r^2 t\\\\\n \u0026amp;= \\frac{22}{7}\\times 14^2 \\times 20\\\\\n \u0026amp;= 12{,}320\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3315em;vertical-align:-2.4158em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9158em;\"\u003e\u003cspan style=\"top:-5.3731em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3917em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4158em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9158em;\"\u003e\u003cspan style=\"top:-5.3731em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3917em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5657em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e320\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4158em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the volume is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e12,320\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n12{,}320\\ \\text{cm}^3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0585em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e320\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h4\u003e\u003cp\u003eA cylindrical drinking container has volume \u003ccode\u003e693 ml\u003c/code\u003e. If its height is \u003ccode\u003e18 cm\u003c/code\u003e and the container is full, determine:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ethe diameter\u003c/li\u003e\n\u003cli\u003ethe total surface area\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eSince \u003ccode\u003e1 ml = 1 cm^3\u003c/code\u003e, we have:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eV\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e693\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nV = 693\\ \\text{cm}^3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e693\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch5 id=\"a-diameter\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-diameter\" class=\"header-mark\" aria-label=\"Direct link to section: a. Diameter\"\u003e\u003c/a\u003ea. Diameter\u003c/h5\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eV\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e693\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.14\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e18\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e693\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e56.52\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e693\u003c/mn\u003e\u003cmn\u003e56.52\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmn\u003e12.26\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmn\u003e3.5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nV \u0026amp;= \\pi r^2 t\\\\\n693 \u0026amp;= 3.14 \\times r^2 \\times 18\\\\\n693 \u0026amp;= 56.52r^2\\\\\nr^2 \u0026amp;= \\frac{693}{56.52}\\\\\nr^2 \u0026amp;\\approx 12.26\\\\\nr \u0026amp;\\approx 3.5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:9.9039em;vertical-align:-4.7019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.2019em;\"\u003e\u003cspan style=\"top:-7.6593em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.22222em;\"\u003eV\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e693\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6111em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e693\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6296em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.7795em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.7205em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7019em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.2019em;\"\u003e\u003cspan style=\"top:-7.6593em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.1352em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.14\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.6111em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e56.52\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.6296em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e56.52\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e693\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.7795em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12.26\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.7205em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7019em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ed\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nd = 2r = 7\\ \\text{cm}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ed\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch5 id=\"b-surface-area\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-surface-area\" class=\"header-mark\" aria-label=\"Direct link to section: b. Surface area\"\u003e\u003c/a\u003eb. Surface area\u003c/h5\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003eS\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e22\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3.5\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3.5\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e18\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e473\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nS \u0026amp;= 2\\pi r(r+t)\\\\\n \u0026amp;= 2 \\times \\frac{22}{7} \\times 3.5 \\times (3.5+18)\\\\\n \u0026amp;= 473\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:5.3074em;vertical-align:-2.4037em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eS\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9037em;\"\u003e\u003cspan style=\"top:-5.3852em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.4037em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5777em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e473\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.4037em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e473\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n473\\ \\text{cm}^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e473\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"important-formulas\" class=\"headerLink\"\u003e\n \u003ca href=\"#important-formulas\" class=\"header-mark\" aria-label=\"Direct link to section: Important Formulas\"\u003e\u003c/a\u003eImportant Formulas\u003c/h2\u003e\u003cul\u003e\n\u003cli\u003ebase area = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi r^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003elateral area = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\pi rt\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003etop area = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi r^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003etotal surface area = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\\pi r(r+t)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003esurface area without top = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi r(r+2t)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003evolume = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eπ\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\pi r^2t\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eπ\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003er\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"geogebra-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#geogebra-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: GeoGebra Simulation\"\u003e\u003c/a\u003eGeoGebra Simulation\u003c/h2\u003e\u003cp\u003eUse the following simulation to explore how changes in radius and height affect the surface area and volume of a cylinder.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/bmy8jqrh/width/650/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Luas Permukaan dan Volume Tabung\" width=\"650px\"\u003e\n \u003c/iframe\u003e\n\u003ch2 id=\"d-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: D. Evaluation\"\u003e\u003c/a\u003eD. Evaluation\u003c/h2\u003e\u003cp\u003eTry the following problems:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eA cylinder has height \u003ccode\u003e14 cm\u003c/code\u003e and base radius \u003ccode\u003e3 cm\u003c/code\u003e. Find its volume.\u003c/li\u003e\n\u003cli\u003eA cylinder is made of sheet metal with radius \u003ccode\u003e14 cm\u003c/code\u003e and height \u003ccode\u003e20 cm\u003c/code\u003e. Find the sheet area needed to make it.\u003c/li\u003e\n\u003cli\u003eA cylindrical water tank has height \u003ccode\u003e2 m\u003c/code\u003e and diameter \u003ccode\u003e7 dm\u003c/code\u003e. The bottom leaks and water flows out at an average rate of \u003ccode\u003e5\u003c/code\u003e liters per minute. If the tank is full, after how many minutes will it be empty?\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThe cylinder is one of the most common curved-surface solids in daily life, appearing in cans, pipes, glasses, and water tanks. Understanding its elements, surface area, and volume makes many applied geometry problems much easier to solve.\u003c/p\u003e\n", "language": "en" }, { "title" : "Surface Area of a Rectangular Prism", "date_published" : "2021-01-04T02:46:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/luas-permukaan-balok/", "url" : "https://www.aantriono.com/en/luas-permukaan-balok/", "summary": " Introduction A. Basic Competencies3.9 Distinguish and determine the surface area and volume of flat-sided solid figures (cube, cuboid, prism, and pyramid). 4.9 Solve problems related to the surface area and volume of flat-sided solid figures (cube, cuboid, prism, and pyramid), as well as their combinations.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"a-basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: A. Basic Competencies\"\u003e\u003c/a\u003eA. Basic Competencies\u003c/h2\u003e\u003cp\u003e3.9 Distinguish and determine the surface area and volume of flat-sided solid figures (cube, cuboid, prism, and pyramid).\n4.9 Solve problems related to the surface area and volume of flat-sided solid figures (cube, cuboid, prism, and pyramid), as well as their combinations.\u003c/p\u003e\n\u003ch2 id=\"b-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: B. Learning Objectives\"\u003e\u003c/a\u003eB. Learning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying the material on the surface area of a cuboid, students are expected to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eFind and determine the surface area of a cuboid.\u003c/li\u003e\n\u003cli\u003eSolve problems related to the surface area of a cuboid.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-material\" class=\"header-mark\" aria-label=\"Direct link to section: C. Material\"\u003e\u003c/a\u003eC. Material\u003c/h2\u003e\u003cp\u003eLook at the following picture!\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhosKddIK7bWMTZazH3yRPiRPahmI4GC7Lofsa_Yf2ylV6ZQY2KPyGh7pq9dyNH4vkO7mZVmKaVNo1e45A7Un32G2Y6f7mmZVZcQfWM22jvGJEx6Pd9E7UVpuEM5s3BEIerBYzA07-1iOIOJu4CLd7bJO4SAYYBhFgY7_2CxAapVom5zmMKoM6M2Xhy/s1280/balok.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"menghitung luas permukaan balok\" border=\"0\" data-original-height=\"400\" data-original-width=\"1280\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhosKddIK7bWMTZazH3yRPiRPahmI4GC7Lofsa_Yf2ylV6ZQY2KPyGh7pq9dyNH4vkO7mZVmKaVNo1e45A7Un32G2Y6f7mmZVZcQfWM22jvGJEx6Pd9E7UVpuEM5s3BEIerBYzA07-1iOIOJu4CLd7bJO4SAYYBhFgY7_2CxAapVom5zmMKoM6M2Xhy/w640-h200/balok.png\" title=\"luas permukaan balok\" width=\"640\" /\u003e\u003c/a\u003e\nThe figure shows a block ABCD.EFGH with length dimensions\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ep\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, width \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003el\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003el\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.01968em;\"\u003el\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003et\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. If we slice the block according to\nthe ribs, it will become a web of beams. One\nThe block net looks like the following picture.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRnf3qIEJ1EfJx1Uj9m3LwZLS1y9HADAcGqHXJfbre-Uq9MLOJ9hgY1rEnGthUJHBktkYnaW0QMIPYTi8JTCc7-iLYOXCopwSXiWa04-vJgn-JCVlK4gjL6RcARptuayVP5KB_4IAL73c/s494/gambar+4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"471\" data-original-width=\"494\" height=\"305\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRnf3qIEJ1EfJx1Uj9m3LwZLS1y9HADAcGqHXJfbre-Uq9MLOJ9hgY1rEnGthUJHBktkYnaW0QMIPYTi8JTCc7-iLYOXCopwSXiWa04-vJgn-JCVlK4gjL6RcARptuayVP5KB_4IAL73c/w320-h305/gambar+4.png\" title=\"Jaring-jaring balok\" width=\"320\" /\u003e\u003c/a\u003e\nThe beam net consists of three pairs of shaped sides\ncongruent rectangles. The surface area of a block is the sum of the areas\nthe three pairs of rectangles on the block.\nThe surface area of ​​the block is formulated as follows:\n\n \n \\(\\begin{aligned}\n\u0026amp;=Area of ​​ABFE \u0026#43; Area of ​​ABCD \u0026#43; Area of ​​ADHE \u0026#43; Area of ​​CDHG \u0026#43; Area of ​​BCGF \u0026#43; Area of ​​EFGH\\\\\n\u0026amp;=(p\\times t)\u0026#43;(p\\times l)\u0026#43;(l \\times t)\u0026#43;(p\\times t)\u0026#43;(l\\times t)\u0026#43;(p\\times l)\\\\\n\u0026amp;=2(pl\u0026#43;pt\u0026#43;lt)\n\\end{aligned}\\)\nProblems example :\nCalculate the surface area of the block if it is known that its length is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e 6 \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, its width\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and the height is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\nSolution:\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003el\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003el\u003c/mi\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e18\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e54\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e108\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin{aligned}\n\u0026amp;L=2 (pl+pt+lt)\\\\\n\u0026amp;L=2((6\\times 3)+(6\\times 4)+(3\\times 4))\\\\\n\u0026amp;L=2(18+24+12)\\\\\n\u0026amp;L=2(54)\\\\\n\u0026amp;L=108\n\\end{aligned}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.5em;vertical-align:-3.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4em;\"\u003e\u003cspan style=\"top:-6em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.84em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.84em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.84em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.84em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.84em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4em;\"\u003e\u003cspan style=\"top:-6.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.01968em;\"\u003epl\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ept\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003elt\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e((\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e))\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e54\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e108\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\nSo, the surface area of ​​the block is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e108\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e108\\ \\text{cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e108\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003ch2 id=\"beam-surface-area-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#beam-surface-area-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: Beam Surface Area Simulation\"\u003e\u003c/a\u003eBeam Surface Area Simulation\u003c/h2\u003e\u003cp\u003eTo better understand the surface area of the beam, please move the following slider to\nright or left.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/xr3c3cz5/width/650/height/400/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Luas Permukaan Balok\" width=\"650px\"\u003e\n\u003c/iframe\u003e\n\u003ch2 id=\"d-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: D. Evaluation\"\u003e\u003c/a\u003eD. Evaluation\u003c/h2\u003e\u003cp\u003eAfter studying the material about the surface area of a cuboid, please answer the following questions to measure your understanding.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eCalculate the surface area of a cuboid with length \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, width \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and height \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm.\u003c/li\u003e\n\u003cli\u003eCalculate the height of the cuboid if its length is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, its width is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e cm, and its surface area is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e208\u003c/mn\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e208\\ \\text{cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e208\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThat concludes the material on the surface area of a cuboid. I hope it is helpful. Thank you 😊🙏.\u003c/p\u003e\n", "language": "en" }, { "title" : "Surface Area of ​​a Cube", "date_published" : "2021-01-03T15:24:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/luas-permukaan-dan-volume-kubus/", "url" : "https://www.aantriono.com/en/luas-permukaan-dan-volume-kubus/", "summary": " Introduction A. Basic Competencies Distinguish and determine area surface and volume of flat sided figures (cube, cuboid, prism and pyramid).4.9 Solve problems related to surface area and volume of a flat-sided figure (cube, cuboid, prism and pyramid), as well as combinations thereof.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"a-basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: A. Basic Competencies\"\u003e\u003c/a\u003eA. Basic Competencies\u003c/h2\u003e\u003col start=\"3\"\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eDistinguish and determine area\nsurface and volume of flat sided figures (cube, cuboid, prism and pyramid).4.9 Solve problems related to surface area and\nvolume of a flat-sided figure (cube, cuboid, prism and pyramid), as well as combinations thereof.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"b-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: B. Learning Objectives\"\u003e\u003c/a\u003eB. Learning Objectives\u003c/h2\u003e\u003cp\u003eAfter\nStudying the surface area of a cube material, students are expected to be able to:1.\nCan find and determine the surface area of a cube.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eGot it\nsolve problems related to the surface area of a cube.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-material\" class=\"header-mark\" aria-label=\"Direct link to section: C. Material\"\u003e\u003c/a\u003eC. Material\u003c/h2\u003e\u003cp\u003eLook at the following picture!\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNMb9dnR2ZVyURArL7MIYhxWefyyjfKJv2YyYFHNY3RVEpJWG-k1dU4qGMREC-aP01ZFyAv0rSm_1mNSrkNGD8afq8KH45GMTkjrcD5Y-gD-wZQE1PGkdp0RI61sZqTg7s3sXUNgck89A_vul8dJ_Xj-u_V70fi2XCqS_qgxzx5ooSxuMc8oo_sILe/s1280/kubus.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"menghitung luas permukaan kubus\" border=\"0\" data-original-height=\"400\" data-original-width=\"1280\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNMb9dnR2ZVyURArL7MIYhxWefyyjfKJv2YyYFHNY3RVEpJWG-k1dU4qGMREC-aP01ZFyAv0rSm_1mNSrkNGD8afq8KH45GMTkjrcD5Y-gD-wZQE1PGkdp0RI61sZqTg7s3sXUNgck89A_vul8dJ_Xj-u_V70fi2XCqS_qgxzx5ooSxuMc8oo_sILe/w640-h200/kubus.png\" title=\"menghitung luas permukaan kubus\" width=\"640\" /\u003e\u003c/a\u003e\nThe figure shows a cube ABCD.EFGH with edge length \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003es\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\nIf we slice the cube according to its edges, it will become\ncube nets. One of the cube nets looks like the picture\nfollowing.\n\u003ca href=\"https://www.blogger.com/#\"\u003e\u003cimg border=\"0\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOtcwyqoF1D-XiLbwUlTO7j-VhIRP08Pj0RDfQ_jtFv-dMto-XoJ7kgMsXI7FbnlX4Qygr5COYmMhG3MiqpYS7c67JF52Bq7Ohw6XWx87JQtnRXuFkBsy3MKgZPmB-1sfPebsFyBuU91M/w242-h320/gambar+2.png\" title=\"Jaring-jaring kubus\" width=\"242\" /\u003e\u003c/a\u003e\nThe cube net consists of six congruent squares\nwith edge size \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003es\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. The surface area of a cube is the sixth sum of areas\nsquare on a cube mesh.\nThen the total surface area of ​​the cube is\n= Area \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABFE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eBFE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e+ Area \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eABCD\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eBC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e+ Area \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003cmi\u003eE\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eADHE\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eDH\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05764em;\"\u003eE\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e+ Area \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eCDHG\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eDH\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eG\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e+ Area \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eBCGF\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eBCGF\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e+ Area\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eE\u003c/mi\u003e\u003cmi\u003eF\u003c/mi\u003e\u003cmi\u003eG\u003c/mi\u003e\u003cmi\u003eH\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eEFGH\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eEFG\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.08125em;\"\u003eH\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin {aligned}\u0026amp;=(s\\times s)+(s\\times s)+(s\\times\ns)+(s\\times s)+(s\\times s)+(s\\times\ns)\\\\\u0026amp;=(s^{2})+(s^{2})+(s^{2})+(s^{2})+(s^{2})+(s^{2})\\\\\u0026amp;=6\\times\ns^{2}\\\\\u0026amp;=6s^{2}\\end{aligned}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0723em;vertical-align:-2.7862em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2862em;\"\u003e\u003cspan style=\"top:-5.3103em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.7862em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.2621em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.7379em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7862em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2862em;\"\u003e\u003cspan style=\"top:-5.4462em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.9221em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8738em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7862em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eProblems example:\nIt is known that the edge length of a cube is 4 cm. Calculate the surface area of the cube\nthat.\nSolution:\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e96\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\begin {aligned} \u0026amp;L=6s^{2}\\\\\u0026amp;L=6\\times 4^{2}\\\\\u0026amp;L=6\\times 16\\\\\u0026amp;L=96\n\\end{aligned}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.0482em;vertical-align:-2.7741em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.2741em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.75em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.75em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.2741em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003es\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.3859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.8859em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e96\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.7741em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\nSo, the surface area of ​​the cube is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e96\u003c/mn\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e96 cm^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e96\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cstrong\u003eCube Surface Area Simulation\u003c/strong\u003e\nTo better understand the surface area of a cube, please carry out the following simulation\nby moving the slider to the right or left.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/vrt7p5ts/width/650/height/400/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Luas Permukaan Kubus\" width=\"650px\"\u003e\n\u003c/iframe\u003e\n\u003ch2 id=\"d-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: D. Evaluation\"\u003e\u003c/a\u003eD. Evaluation\u003c/h2\u003e\u003cp\u003eTo better understand the surface area of a cube, please do an evaluation\nnext!\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eCalculate the surface area of ​​a cube with an edge length of 12 cm.\u003c/li\u003e\n\u003cli\u003eIt is known that the total length of the edges of a cube is 108 cm. Determine the area\nthe surface of the cube.\u003c/li\u003e\n\u003cli\u003eIt is known that the surface area of ​​a cube is 726 \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmtext\u003ecm\u003c/mtext\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\text{cm}^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ecm\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Determine the length of the rib\nthe cube.\u003c/li\u003e\n\u003cli\u003eSatrio will make 10 identical cube frames using wire. If the edge length of the cube is 12 cm, what is the minimum wire length required?\nThat\u0026rsquo;s the material about the surface area of ​​a cube. I hope this is helpful.\nlove.😊🙏\u003c/li\u003e\n\u003c/ol\u003e\n", "language": "en" }, { "title" : "Overview of GeoGebra", "date_published" : "2020-12-05T14:20:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sekilas-tentang-geogebra/", "url" : "https://www.aantriono.com/en/sekilas-tentang-geogebra/", "summary": " IntroductionGeoGebra is a widely used mathematics application for learning geometry, algebra, graphing, statistics, and symbolic computation. It is popular among students, teachers, and lecturers because it makes mathematical ideas easier to visualize and explore interactively.\nA Brief History of GeoGebraGeoGebra was developed by Markus Hohenwarter starting in 2001. He is an Austrian mathematician and professor at Johannes Kepler University Linz. During his university studies in computer science and applied mathematics, he developed GeoGebra as a mathematics education tool, and it later received many software awards in Europe and the United States.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eGeoGebra is a widely used mathematics application for learning geometry, algebra, graphing, statistics, and symbolic computation. It is popular among students, teachers, and lecturers because it makes mathematical ideas easier to visualize and explore interactively.\u003c/p\u003e\n\u003ch3 id=\"a-brief-history-of-geogebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-brief-history-of-geogebra\" class=\"header-mark\" aria-label=\"Direct link to section: A Brief History of GeoGebra\"\u003e\u003c/a\u003eA Brief History of GeoGebra\u003c/h3\u003e\u003cp\u003eGeoGebra was developed by Markus Hohenwarter starting in 2001. He is an Austrian mathematician and professor at Johannes Kepler University Linz. During his university studies in computer science and applied mathematics, he developed GeoGebra as a mathematics education tool, and it later received many software awards in Europe and the United States.\u003c/p\u003e\n\u003cp\u003eAfter completing his dissertation at the University of Salzburg in 2006, Hohenwarter also worked on teacher training projects in Austria, England, and the United States. His research has focused on the use of technology in mathematics education.\u003c/p\u003e\n\u003cp\u003eAccording to Hohenwarter, GeoGebra is a computer program for teaching mathematics, especially geometry and algebra. It is freely available and can be downloaded from \u003ca href=\"https://www.geogebra.org/\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ewww.geogebra.org\u003c/a\u003e.\u003c/p\u003e\n\u003ch3 id=\"benefits-of-geogebra-in-mathematics-learning\" class=\"headerLink\"\u003e\n \u003ca href=\"#benefits-of-geogebra-in-mathematics-learning\" class=\"header-mark\" aria-label=\"Direct link to section: Benefits of GeoGebra in Mathematics Learning\"\u003e\u003c/a\u003eBenefits of GeoGebra in Mathematics Learning\u003c/h3\u003e\u003cp\u003eSome of the main benefits are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eIt can produce geometric constructions quickly and accurately, including complex ones.\u003c/li\u003e\n\u003cli\u003eIts animation and manipulation features provide strong visual support for understanding geometry.\u003c/li\u003e\n\u003cli\u003eIt can be used as a feedback tool to verify whether a construction is correct.\u003c/li\u003e\n\u003cli\u003eIt helps users investigate and demonstrate properties of geometric objects.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eGeoGebra has continued to develop over time. Newer versions improve older ones, and three-dimensional graphics became much more accessible beginning with later releases. In many current tutorials, GeoGebra 6 is the standard version being used.\u003c/p\u003e\n\u003ch3 id=\"geogebra-interface-anatomy\" class=\"headerLink\"\u003e\n \u003ca href=\"#geogebra-interface-anatomy\" class=\"header-mark\" aria-label=\"Direct link to section: GeoGebra Interface Anatomy\"\u003e\u003c/a\u003eGeoGebra Interface Anatomy\u003c/h3\u003e\u003cp\u003eWhen you first open GeoGebra, the screen looks roughly like this.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp1fqXzzY7u7ohSmX36YbAvGJi_aBoVeTC0gKU01hRhEeLIs8SnQUiNZIxRi7LBvYEVgQkyliT3BnM2ZqUHBaCruErrK0bCtqx_BMy9LVfVM-SoPA8jOH7u3dVoVB9GLa5tw-Rq7WxP9iFL1Dg7uDVydnqv7A7S2XFmMFP0ln4MpLZfkb41fGyb_Tf3nI/w400-h193/Screenshot%20from%202023-09-24%2020-12-13.png\" title=\"Initial GeoGebra interface\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp1fqXzzY7u7ohSmX36YbAvGJi_aBoVeTC0gKU01hRhEeLIs8SnQUiNZIxRi7LBvYEVgQkyliT3BnM2ZqUHBaCruErrK0bCtqx_BMy9LVfVM-SoPA8jOH7u3dVoVB9GLa5tw-Rq7WxP9iFL1Dg7uDVydnqv7A7S2XFmMFP0ln4MpLZfkb41fGyb_Tf3nI/w400-h193/Screenshot%20from%202023-09-24%2020-12-13.png\" data-sub-html=\"\u003ch2\u003eInitial GeoGebra interface\u003c/h2\u003e\u003cp\u003eInitial GeoGebra interface\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp1fqXzzY7u7ohSmX36YbAvGJi_aBoVeTC0gKU01hRhEeLIs8SnQUiNZIxRi7LBvYEVgQkyliT3BnM2ZqUHBaCruErrK0bCtqx_BMy9LVfVM-SoPA8jOH7u3dVoVB9GLa5tw-Rq7WxP9iFL1Dg7uDVydnqv7A7S2XFmMFP0ln4MpLZfkb41fGyb_Tf3nI/w400-h193/Screenshot%20from%202023-09-24%2020-12-13.png' alt=\"Initial GeoGebra interface\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eInitial GeoGebra interface\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp1fqXzzY7u7ohSmX36YbAvGJi_aBoVeTC0gKU01hRhEeLIs8SnQUiNZIxRi7LBvYEVgQkyliT3BnM2ZqUHBaCruErrK0bCtqx_BMy9LVfVM-SoPA8jOH7u3dVoVB9GLa5tw-Rq7WxP9iFL1Dg7uDVydnqv7A7S2XFmMFP0ln4MpLZfkb41fGyb_Tf3nI/s1914/Screenshot%20from%202023-09-24%2020-12-13.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp1fqXzzY7u7ohSmX36YbAvGJi_aBoVeTC0gKU01hRhEeLIs8SnQUiNZIxRi7LBvYEVgQkyliT3BnM2ZqUHBaCruErrK0bCtqx_BMy9LVfVM-SoPA8jOH7u3dVoVB9GLa5tw-Rq7WxP9iFL1Dg7uDVydnqv7A7S2XFmMFP0ln4MpLZfkb41fGyb_Tf3nI/s1914/Screenshot%20from%202023-09-24%2020-12-13.png\u003c/a\u003e)\u003c/p\u003e\n\u003cp\u003eOn the right side, there is usually a \u003cstrong\u003ePerspectives\u003c/strong\u003e panel. This panel lets you choose different workspace layouts. In general, GeoGebra offers six main views.\u003c/p\u003e\n\u003ch4 id=\"1-algebra\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-algebra\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Algebra\"\u003e\u003c/a\u003e1. Algebra\u003c/h4\u003e\u003cp\u003eThis view combines an algebra panel and a graphics panel.\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eThe left side shows algebraic forms of objects or equations.\u003c/li\u003e\n\u003cli\u003eThe right side shows their graphs or geometric representations.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch4 id=\"2-geometry\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-geometry\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Geometry\"\u003e\u003c/a\u003e2. Geometry\u003c/h4\u003e\u003cp\u003eThis view focuses on the geometric representation of objects without emphasizing the algebra panel.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjulmOXppDKVh6SfOXR35_tce5cP1WYlHz6netz7MhD6MvAEoaTH712U_qoCV4RkEZeCpFGN2scJJ3BjMgDcURfsNJJ48Q1ca6-Kw99Hy4KRn-4c6dxwAi_dufx5V1_LiMXDdFED-aj7kzcJTMb2VzGA0XoMCW1lHUqvz4nUa1yjZfB8u8GInxgzgPUBmQ/w400-h225/Screenshot%20from%202023-09-24%2020-17-31.png\" title=\"GeoGebra Geometry view\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjulmOXppDKVh6SfOXR35_tce5cP1WYlHz6netz7MhD6MvAEoaTH712U_qoCV4RkEZeCpFGN2scJJ3BjMgDcURfsNJJ48Q1ca6-Kw99Hy4KRn-4c6dxwAi_dufx5V1_LiMXDdFED-aj7kzcJTMb2VzGA0XoMCW1lHUqvz4nUa1yjZfB8u8GInxgzgPUBmQ/w400-h225/Screenshot%20from%202023-09-24%2020-17-31.png\" data-sub-html=\"\u003ch2\u003eGeoGebra Geometry view\u003c/h2\u003e\u003cp\u003eGeoGebra Geometry view\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjulmOXppDKVh6SfOXR35_tce5cP1WYlHz6netz7MhD6MvAEoaTH712U_qoCV4RkEZeCpFGN2scJJ3BjMgDcURfsNJJ48Q1ca6-Kw99Hy4KRn-4c6dxwAi_dufx5V1_LiMXDdFED-aj7kzcJTMb2VzGA0XoMCW1lHUqvz4nUa1yjZfB8u8GInxgzgPUBmQ/w400-h225/Screenshot%20from%202023-09-24%2020-17-31.png' alt=\"GeoGebra Geometry view\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eGeoGebra Geometry view\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjulmOXppDKVh6SfOXR35_tce5cP1WYlHz6netz7MhD6MvAEoaTH712U_qoCV4RkEZeCpFGN2scJJ3BjMgDcURfsNJJ48Q1ca6-Kw99Hy4KRn-4c6dxwAi_dufx5V1_LiMXDdFED-aj7kzcJTMb2VzGA0XoMCW1lHUqvz4nUa1yjZfB8u8GInxgzgPUBmQ/s1547/Screenshot%20from%202023-09-24%2020-17-31.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjulmOXppDKVh6SfOXR35_tce5cP1WYlHz6netz7MhD6MvAEoaTH712U_qoCV4RkEZeCpFGN2scJJ3BjMgDcURfsNJJ48Q1ca6-Kw99Hy4KRn-4c6dxwAi_dufx5V1_LiMXDdFED-aj7kzcJTMb2VzGA0XoMCW1lHUqvz4nUa1yjZfB8u8GInxgzgPUBmQ/s1547/Screenshot%20from%202023-09-24%2020-17-31.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch4 id=\"3-spreadsheet\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-spreadsheet\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Spreadsheet\"\u003e\u003c/a\u003e3. Spreadsheet\u003c/h4\u003e\u003cp\u003eThe spreadsheet view is organized in rows and columns. It can be used for matrices, data tables, and other mathematical objects arranged like a worksheet.\u003c/p\u003e\n\u003cp\u003eYou can enter not only numbers, but also points, functions, and commands. When possible, GeoGebra immediately shows their graphical representation in the graphics view as well.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXT9xf0uXPqXGRNyjgL2xiL4yCFjaRJ_etFn3VJCz4CIm_J12RDCLrzG5im4YVnsGFb2rxUEnJRBQRBrs3PLubSLpiPE7H4DlgEXu4Ihn0Vawn4cbEUSL4ZTRGws4-a_whRf8yvKbpQtkm72ErEzeWYxeejL01XdzeqjKRD-6bxWTJwTXB35li4c0pkHY/w400-h191/Screenshot%20from%202023-09-24%2020-19-55.png\" title=\"GeoGebra Spreadsheet view\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXT9xf0uXPqXGRNyjgL2xiL4yCFjaRJ_etFn3VJCz4CIm_J12RDCLrzG5im4YVnsGFb2rxUEnJRBQRBrs3PLubSLpiPE7H4DlgEXu4Ihn0Vawn4cbEUSL4ZTRGws4-a_whRf8yvKbpQtkm72ErEzeWYxeejL01XdzeqjKRD-6bxWTJwTXB35li4c0pkHY/w400-h191/Screenshot%20from%202023-09-24%2020-19-55.png\" data-sub-html=\"\u003ch2\u003eGeoGebra Spreadsheet view\u003c/h2\u003e\u003cp\u003eGeoGebra Spreadsheet view\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXT9xf0uXPqXGRNyjgL2xiL4yCFjaRJ_etFn3VJCz4CIm_J12RDCLrzG5im4YVnsGFb2rxUEnJRBQRBrs3PLubSLpiPE7H4DlgEXu4Ihn0Vawn4cbEUSL4ZTRGws4-a_whRf8yvKbpQtkm72ErEzeWYxeejL01XdzeqjKRD-6bxWTJwTXB35li4c0pkHY/w400-h191/Screenshot%20from%202023-09-24%2020-19-55.png' alt=\"GeoGebra Spreadsheet view\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eGeoGebra Spreadsheet view\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXT9xf0uXPqXGRNyjgL2xiL4yCFjaRJ_etFn3VJCz4CIm_J12RDCLrzG5im4YVnsGFb2rxUEnJRBQRBrs3PLubSLpiPE7H4DlgEXu4Ihn0Vawn4cbEUSL4ZTRGws4-a_whRf8yvKbpQtkm72ErEzeWYxeejL01XdzeqjKRD-6bxWTJwTXB35li4c0pkHY/s1916/Screenshot%20from%202023-09-24%2020-19-55.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXT9xf0uXPqXGRNyjgL2xiL4yCFjaRJ_etFn3VJCz4CIm_J12RDCLrzG5im4YVnsGFb2rxUEnJRBQRBrs3PLubSLpiPE7H4DlgEXu4Ihn0Vawn4cbEUSL4ZTRGws4-a_whRf8yvKbpQtkm72ErEzeWYxeejL01XdzeqjKRD-6bxWTJwTXB35li4c0pkHY/s1916/Screenshot%20from%202023-09-24%2020-19-55.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch4 id=\"4-cas\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-cas\" class=\"header-mark\" aria-label=\"Direct link to section: 4. CAS\"\u003e\u003c/a\u003e4. CAS\u003c/h4\u003e\u003cp\u003eCAS stands for \u003cstrong\u003eComputer Algebra System\u003c/strong\u003e. This view is used for symbolic computation, such as simplifying expressions, solving equations, or working with derivatives and integrals symbolically.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMksNG-9FcniTZl4HDQI46ryF94fGQbtamP3evjSAhhViq-wQ9JWuzkCeTErADsyDFzcFLdOIZDexvmzyjnFlAoTVW47Y5rh4wMLVdursb-Aa89laOEBQIRtKfZRZuPWYM8xSLQoNHJNZRueQbnpSdQUu_GFGLeyC_-7wGEkKrWrfEUVGT7wBPNGDRJJg/w400-h191/Screenshot%20from%202023-09-24%2020-25-07.png\" title=\"GeoGebra CAS view\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMksNG-9FcniTZl4HDQI46ryF94fGQbtamP3evjSAhhViq-wQ9JWuzkCeTErADsyDFzcFLdOIZDexvmzyjnFlAoTVW47Y5rh4wMLVdursb-Aa89laOEBQIRtKfZRZuPWYM8xSLQoNHJNZRueQbnpSdQUu_GFGLeyC_-7wGEkKrWrfEUVGT7wBPNGDRJJg/w400-h191/Screenshot%20from%202023-09-24%2020-25-07.png\" data-sub-html=\"\u003ch2\u003eGeoGebra CAS view\u003c/h2\u003e\u003cp\u003eGeoGebra CAS view\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMksNG-9FcniTZl4HDQI46ryF94fGQbtamP3evjSAhhViq-wQ9JWuzkCeTErADsyDFzcFLdOIZDexvmzyjnFlAoTVW47Y5rh4wMLVdursb-Aa89laOEBQIRtKfZRZuPWYM8xSLQoNHJNZRueQbnpSdQUu_GFGLeyC_-7wGEkKrWrfEUVGT7wBPNGDRJJg/w400-h191/Screenshot%20from%202023-09-24%2020-25-07.png' alt=\"GeoGebra CAS view\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eGeoGebra CAS view\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMksNG-9FcniTZl4HDQI46ryF94fGQbtamP3evjSAhhViq-wQ9JWuzkCeTErADsyDFzcFLdOIZDexvmzyjnFlAoTVW47Y5rh4wMLVdursb-Aa89laOEBQIRtKfZRZuPWYM8xSLQoNHJNZRueQbnpSdQUu_GFGLeyC_-7wGEkKrWrfEUVGT7wBPNGDRJJg/s1916/Screenshot%20from%202023-09-24%2020-25-07.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMksNG-9FcniTZl4HDQI46ryF94fGQbtamP3evjSAhhViq-wQ9JWuzkCeTErADsyDFzcFLdOIZDexvmzyjnFlAoTVW47Y5rh4wMLVdursb-Aa89laOEBQIRtKfZRZuPWYM8xSLQoNHJNZRueQbnpSdQUu_GFGLeyC_-7wGEkKrWrfEUVGT7wBPNGDRJJg/s1916/Screenshot%20from%202023-09-24%2020-25-07.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch4 id=\"5-3d-graphics\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-3d-graphics\" class=\"header-mark\" aria-label=\"Direct link to section: 5. 3D Graphics\"\u003e\u003c/a\u003e5. 3D Graphics\u003c/h4\u003e\u003cp\u003eThis view is designed for three-dimensional objects. Its structure is similar to the algebra-plus-graphics layout, but the graphics panel is used for 3D visualization.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvNDFeD6nZwbzjHJrWVRmIbOx7drtGln7vk4RcNcK8MLTSTXj3Yw8i5kGiG6JB0ZgTiyTpps8rZfxm4crn3PXprLrkP4CbXiPia1k7AtqxDk62VzaexjLH78BAut-mWt8SJD9H5oSvINA9zEFmgs3QmgeBn8Ve3oaGjlLo3Q_uhNCqTeTSKYHBF_1cyy8/w400-h191/Screenshot%20from%202023-09-24%2020-26-39.png\" title=\"GeoGebra 3D view\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvNDFeD6nZwbzjHJrWVRmIbOx7drtGln7vk4RcNcK8MLTSTXj3Yw8i5kGiG6JB0ZgTiyTpps8rZfxm4crn3PXprLrkP4CbXiPia1k7AtqxDk62VzaexjLH78BAut-mWt8SJD9H5oSvINA9zEFmgs3QmgeBn8Ve3oaGjlLo3Q_uhNCqTeTSKYHBF_1cyy8/w400-h191/Screenshot%20from%202023-09-24%2020-26-39.png\" data-sub-html=\"\u003ch2\u003eGeoGebra 3D view\u003c/h2\u003e\u003cp\u003eGeoGebra 3D view\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvNDFeD6nZwbzjHJrWVRmIbOx7drtGln7vk4RcNcK8MLTSTXj3Yw8i5kGiG6JB0ZgTiyTpps8rZfxm4crn3PXprLrkP4CbXiPia1k7AtqxDk62VzaexjLH78BAut-mWt8SJD9H5oSvINA9zEFmgs3QmgeBn8Ve3oaGjlLo3Q_uhNCqTeTSKYHBF_1cyy8/w400-h191/Screenshot%20from%202023-09-24%2020-26-39.png' alt=\"GeoGebra 3D view\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eGeoGebra 3D view\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvNDFeD6nZwbzjHJrWVRmIbOx7drtGln7vk4RcNcK8MLTSTXj3Yw8i5kGiG6JB0ZgTiyTpps8rZfxm4crn3PXprLrkP4CbXiPia1k7AtqxDk62VzaexjLH78BAut-mWt8SJD9H5oSvINA9zEFmgs3QmgeBn8Ve3oaGjlLo3Q_uhNCqTeTSKYHBF_1cyy8/s1916/Screenshot%20from%202023-09-24%2020-26-39.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvNDFeD6nZwbzjHJrWVRmIbOx7drtGln7vk4RcNcK8MLTSTXj3Yw8i5kGiG6JB0ZgTiyTpps8rZfxm4crn3PXprLrkP4CbXiPia1k7AtqxDk62VzaexjLH78BAut-mWt8SJD9H5oSvINA9zEFmgs3QmgeBn8Ve3oaGjlLo3Q_uhNCqTeTSKYHBF_1cyy8/s1916/Screenshot%20from%202023-09-24%2020-26-39.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch4 id=\"6-probability\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-probability\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Probability\"\u003e\u003c/a\u003e6. Probability\u003c/h4\u003e\u003cp\u003eThis view is used for probability and statistics. It helps users inspect statistical distributions and carry out related calculations.\u003c/p\u003e\n\u003cp\u003e[\u003c/p\u003e\n\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglBp-OpXievl-DpBm0gbwT5cRhfG3ZA0q5U1VjwvB4-NXBBLpLQxPIPVT9CyoDPY2v18Q68c5HKppauQsKiTnCB8KJGv26QAUsjQ5cftS1reU13qBSOjT39fQnF2Z4DNhJ48JD1-8_Km2v61cQ3N4b1O4eG-_jov04-DFqXX1gDF4rpxY9DqpdpTs6QQI/w400-h191/Screenshot%20from%202023-09-24%2020-22-35.png\" title=\"GeoGebra Probability view\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglBp-OpXievl-DpBm0gbwT5cRhfG3ZA0q5U1VjwvB4-NXBBLpLQxPIPVT9CyoDPY2v18Q68c5HKppauQsKiTnCB8KJGv26QAUsjQ5cftS1reU13qBSOjT39fQnF2Z4DNhJ48JD1-8_Km2v61cQ3N4b1O4eG-_jov04-DFqXX1gDF4rpxY9DqpdpTs6QQI/w400-h191/Screenshot%20from%202023-09-24%2020-22-35.png\" data-sub-html=\"\u003ch2\u003eGeoGebra Probability view\u003c/h2\u003e\u003cp\u003eGeoGebra Probability view\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglBp-OpXievl-DpBm0gbwT5cRhfG3ZA0q5U1VjwvB4-NXBBLpLQxPIPVT9CyoDPY2v18Q68c5HKppauQsKiTnCB8KJGv26QAUsjQ5cftS1reU13qBSOjT39fQnF2Z4DNhJ48JD1-8_Km2v61cQ3N4b1O4eG-_jov04-DFqXX1gDF4rpxY9DqpdpTs6QQI/w400-h191/Screenshot%20from%202023-09-24%2020-22-35.png' alt=\"GeoGebra Probability view\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eGeoGebra Probability view\u003c/figcaption\u003e\n\u003c/figure\u003e](\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglBp-OpXievl-DpBm0gbwT5cRhfG3ZA0q5U1VjwvB4-NXBBLpLQxPIPVT9CyoDPY2v18Q68c5HKppauQsKiTnCB8KJGv26QAUsjQ5cftS1reU13qBSOjT39fQnF2Z4DNhJ48JD1-8_Km2v61cQ3N4b1O4eG-_jov04-DFqXX1gDF4rpxY9DqpdpTs6QQI/s1916/Screenshot%20from%202023-09-24%2020-22-35.png\" target=\"_blank\" rel=\"noopener noreferrer\"\u003ehttps://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglBp-OpXievl-DpBm0gbwT5cRhfG3ZA0q5U1VjwvB4-NXBBLpLQxPIPVT9CyoDPY2v18Q68c5HKppauQsKiTnCB8KJGv26QAUsjQ5cftS1reU13qBSOjT39fQnF2Z4DNhJ48JD1-8_Km2v61cQ3N4b1O4eG-_jov04-DFqXX1gDF4rpxY9DqpdpTs6QQI/s1916/Screenshot%20from%202023-09-24%2020-22-35.png\u003c/a\u003e)\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eGeoGebra is a very useful tool for learning and teaching mathematics because it combines visualization, computation, and interactivity in one application. Once you understand its background and its main interface views, it becomes much easier to follow more advanced GeoGebra tutorials.\u003c/p\u003e\n", "language": "en" }, { "title" : "Systems of Linear Equations in Two Variables", "date_published" : "2020-10-25T03:54:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sistem-persamaan-linear-dua-variabel/", "url" : "https://www.aantriono.com/en/sistem-persamaan-linear-dua-variabel/", "summary": " Introduction\nBasic Competencies3.5 Explain systems of linear equations in two variables and their solutions in contextual problems.\n4.5 Solve problems related to systems of linear equations in two variables.\nLearning ObjectivesAfter studying this material, students are expected to be able to:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQYlXvMoMVjXO1IafzR1QxgZtpVZMB8nwuL0655-80spvNihXslZImwRqZV8Ny-JT6Zod5y1wPf1N0KtfNpfgiQl4Mgp970Cvk4q5ed-itGe47OR1hUR8PZnjpL6N2iMLBz7QStt7G9aQ/w640-h314/SPLDV.jpg' alt=\"Systems of linear equations in two variables\" \u003e\u003c/p\u003e\n\u003ch3 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h3\u003e\u003cp\u003e3.5 Explain systems of linear equations in two variables and their solutions in contextual problems.\u003c/p\u003e\n\u003cp\u003e4.5 Solve problems related to systems of linear equations in two variables.\u003c/p\u003e\n\u003ch3 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h3\u003e\u003cp\u003eAfter studying this material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eexplain the meaning of a linear equation in two variables,\u003c/li\u003e\n\u003cli\u003eexplain the general form of a system of linear equations in two variables,\u003c/li\u003e\n\u003cli\u003esolve SLETV problems using graph, substitution, elimination, and mixed methods,\u003c/li\u003e\n\u003cli\u003esolve word problems related to systems of linear equations in two variables.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h2\u003e\u003ch3 id=\"1-linear-equations-in-two-variables\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-linear-equations-in-two-variables\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Linear Equations in Two Variables\"\u003e\u003c/a\u003e1. Linear Equations in Two Variables\u003c/h3\u003e\u003cp\u003eObserve the following equations.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x + 2y = 1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{m}{n} - \\dfrac{n}{2} = 5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.7936em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5p + 6q = -20\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ep\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eq\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eEach equation has two variables, and each variable has degree one. Therefore, they are examples of \u003cstrong\u003elinear equations in two variables\u003c/strong\u003e.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eNote\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eA linear equation in two variables can be written in the general form\n\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eax + by = c\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi mathvariant=\"double-struck\"\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea,b,c \\in \\mathbb{R}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6889em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathbb\"\u003eR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\neq 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb \\neq 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThe solution of a linear equation in two variables can be found by inspection or by algebraic operations.\u003c/p\u003e\n\u003ch4 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eDetermine the non-negative integer solutions \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x,y)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e that satisfy:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n4x + 3y = 12\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n4x + 3y \u0026amp;= 12 \\\\\n0 + 3y \u0026amp;= 12 \\\\\n3y \u0026amp;= 12 \\\\\ny \u0026amp;= 4\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" 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style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan 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displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n4x + 3y \u0026amp;= 12 \\\\\n4(3) + 3y \u0026amp;= 12 \\\\\n12 + 3y \u0026amp;= 12 \\\\\n3y \u0026amp;= 0 \\\\\ny \u0026amp;= 0\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan 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class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the solution set is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\{(0,4),(3,0)\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-systems-of-linear-equations-in-two-variables\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-systems-of-linear-equations-in-two-variables\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Systems of Linear Equations in Two Variables\"\u003e\u003c/a\u003e2. Systems of Linear Equations in Two Variables\u003c/h3\u003e\u003cp\u003eA system of linear equations in two variables is a collection of two or more linear equations that use the same variables.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cstrong\u003eGeneral form\u003c/strong\u003e\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\u0026gt; \\left\\{\n\u0026gt; \\begin{aligned}\n\u0026gt; a_1x+b_1y\u0026amp;=c_1 \\\\\n\u0026gt; a_2x+b_2y\u0026amp;=c_2\n\u0026gt; \\end{aligned}\n\u0026gt; \\right.\n\u0026gt; \u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\n\u003c/blockquote\u003e\u003cp\u003eExamples:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\left\\{\\begin{aligned}2x+y\u0026amp;=-5\\\\3x+5y\u0026amp;=-4\\end{aligned}\\right.\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\left\\{\\begin{aligned}y\u0026amp;=3x-2\\\\2x+y\u0026amp;=8\\end{aligned}\\right.\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"solving-sletv\" class=\"headerLink\"\u003e\n \u003ca href=\"#solving-sletv\" class=\"header-mark\" aria-label=\"Direct link to section: Solving SLETV\"\u003e\u003c/a\u003eSolving SLETV\u003c/h2\u003e\u003cp\u003eSystems of linear equations in two variables can be solved by:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003egraph method,\u003c/li\u003e\n\u003cli\u003esubstitution method,\u003c/li\u003e\n\u003cli\u003eelimination method,\u003c/li\u003e\n\u003cli\u003emixed method.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch3 id=\"3-graph-method\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-graph-method\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Graph Method\"\u003e\u003c/a\u003e3. Graph Method\u003c/h3\u003e\u003cp\u003eIn the graph method, the solution is the intersection point of the two lines on the Cartesian plane.\u003c/p\u003e\n\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eFind the solution set of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left\\{\n\\begin{aligned}\nx+y\u0026amp;=5 \\\\\nx-y\u0026amp;=1\n\\end{aligned}\n\\right.\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFor the equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex+y=5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003ex\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e0\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e5\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003ey\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e5\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e0\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e(x, y)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e(0, 5)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e(5, 0)\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eFor the equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex-y=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\n\u003cdiv class=\"table-wrapper\"\u003e\n \u003ctable\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth style=\"text-align: \"\u003ex\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e0\u003c/th\u003e\n \u003cth style=\"text-align: \"\u003e1\u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003ey\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e-1\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e0\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"text-align: \"\u003e(x, y)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e(0, -1)\u003c/td\u003e\n \u003ctd style=\"text-align: \"\u003e(1, 0)\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eThe graph is as follows.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhly_1r-GLlXIWaAU22n1YCrSxpoe654t-aN5hJvTX3HUWQUwjjrxYkPVxCFunkbMutiH87xL0ggu3toddu-EYUnauUwDwa3HiVVJnPkMu8odYJS7voNN6lYxjULw8emBnemHgniSaEGdM/s463/spldv1.png' alt=\"SLETV graph\" \u003e\u003c/p\u003e\n\u003cp\u003eThe two lines intersect at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(3,2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Therefore, the solution set is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\{(3,2)\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"4-substitution-method\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-substitution-method\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Substitution Method\"\u003e\u003c/a\u003e4. Substitution Method\u003c/h3\u003e\u003cp\u003eThe substitution method is done by expressing one variable in terms of the other, then substituting it into the second equation.\u003c/p\u003e\n\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eSolve:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left\\{\n\\begin{aligned}\n5x+5y\u0026amp;=25 \\qquad (1)\\\\\n3x+6y\u0026amp;=24 \\qquad (2)\n\\end{aligned}\n\\right.\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFrom equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n5x+5y \u0026amp;= 25 \\\\\n5y \u0026amp;= 25-5x \\\\\ny \u0026amp;= 5-x\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSubstitute \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey=5-x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e into equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n3x+6y \u0026amp;= 24 \\\\\n3x+6(5-x) \u0026amp;= 24 \\\\\n3x+30-6x \u0026amp;= 24 \\\\\n-3x \u0026amp;= -6 \\\\\nx \u0026amp;= 2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.5em;vertical-align:-3.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4em;\"\u003e\u003cspan style=\"top:-6.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4em;\"\u003e\u003cspan style=\"top:-6.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSubstitute \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e into equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n5(2)+5y \u0026amp;= 25 \\\\\n10+5y \u0026amp;= 25 \\\\\n5y \u0026amp;= 15 \\\\\ny \u0026amp;= 3\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the solution set is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\{(2,3)\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"5-elimination-method\" class=\"headerLink\"\u003e\n \u003ca href=\"#5-elimination-method\" class=\"header-mark\" aria-label=\"Direct link to section: 5. Elimination Method\"\u003e\u003c/a\u003e5. Elimination Method\u003c/h3\u003e\u003cp\u003eThe elimination method removes one variable by making its coefficients equal first if necessary.\u003c/p\u003e\n\u003ch4 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eSolve:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left\\{\n\\begin{aligned}\n2x+y\u0026amp;=8 \\\\\nx-y\u0026amp;=10\n\\end{aligned}\n\\right.\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAdd the equations to eliminate \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\" rowlines=\"none solid\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e18\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n2x+y \u0026amp;= 8 \\\\\nx-y \u0026amp;= 10 \\\\\n\\hline\n3x \u0026amp;= 18\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"hline\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eHence:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eNow eliminate \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e by multiplying the second equation by 2:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\" rowlines=\"none solid\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n2x+y \u0026amp;= 8 \\\\\n2x-2y \u0026amp;= 20 \\\\\n\\hline\n3y \u0026amp;= -12\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"hline\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny=-4\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the solution set is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\{(6,-4)\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"6-mixed-method\" class=\"headerLink\"\u003e\n \u003ca href=\"#6-mixed-method\" class=\"header-mark\" aria-label=\"Direct link to section: 6. Mixed Method\"\u003e\u003c/a\u003e6. Mixed Method\u003c/h3\u003e\u003cp\u003eThe mixed method combines elimination and substitution.\u003c/p\u003e\n\u003ch4 id=\"example-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-4\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eSolve:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left\\{\n\\begin{aligned}\n2x+y\u0026amp;=5 \\qquad (1)\\\\\n3x-2y\u0026amp;=11 \\qquad (2)\n\\end{aligned}\n\\right.\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eMake the coefficients of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e equal:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e22\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n6x+3y \u0026amp;= 15 \\\\\n6x-4y \u0026amp;= 22\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSubtract the equations:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\" rowlines=\"none solid\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e22\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n6x+3y \u0026amp;= 15 \\\\\n6x-4y \u0026amp;= 22 \\\\\n\\hline\n7y \u0026amp;= -7\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e22\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"hline\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThus:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny=-1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSubstitute into equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n2x+y \u0026amp;= 5 \\\\\n2x-1 \u0026amp;= 5 \\\\\n2x \u0026amp;= 6 \\\\\nx \u0026amp;= 3\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the solution set is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\{(3,-1)\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"7-word-problem\" class=\"headerLink\"\u003e\n \u003ca href=\"#7-word-problem\" class=\"header-mark\" aria-label=\"Direct link to section: 7. Word Problem\"\u003e\u003c/a\u003e7. Word Problem\u003c/h3\u003e\u003cp\u003eWord problems involving SLETV are solved by first creating a mathematical model and then solving the resulting system.\u003c/p\u003e\n\u003ch4 id=\"example-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-5\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eThe price of five tables and eight chairs is Rp1,150,000.00, while the price of three tables and five chairs is Rp700,000.00. Determine the price of each table and chair.\u003c/p\u003e\n\u003cp\u003eLet:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003etable price = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003echair price = \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThen:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e{\u003c/mo\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1.150.000\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e700.000\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left\\{\n\\begin{aligned}\n5x+8y\u0026amp;=1.150.000 \\qquad (1)\\\\\n3x+5y\u0026amp;=700.000 \\qquad (2)\n\\end{aligned}\n\\right.\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size4\"\u003e{\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1.150.000\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e700.000\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eMultiply equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e by 3 and equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e by 5:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.450.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.500.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n15x+24y \u0026amp;= 3.450.000 \\\\\n15x+25y \u0026amp;= 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.450.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.500.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSubtract:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\" rowlines=\"none solid\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e25\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.500.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e24\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3.450.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e50.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n15x+25y \u0026amp;= 3.500.000 \\\\\n15x+24y \u0026amp;= 3.450.000 \\\\\n\\hline\ny \u0026amp;= 50.000\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.5em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e25\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e24\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.500.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3.450.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e50.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:4.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"hline\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSubstitute \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e50,000\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey=50{,}000\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e50\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e into equation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e700.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e50.000\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e700.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e250.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e700.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e450.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e150.000\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n3x+5y \u0026amp;= 700.000 \\\\\n3x+5(50.000) \u0026amp;= 700.000 \\\\\n3x+250.000 \u0026amp;= 700.000 \\\\\n3x \u0026amp;= 450.000 \\\\\nx \u0026amp;= 150.000\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.5em;vertical-align:-3.5em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4em;\"\u003e\u003cspan style=\"top:-6.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e50.000\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e250.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4em;\"\u003e\u003cspan style=\"top:-6.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e700.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e700.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e700.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e450.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e150.000\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003etable price = Rp150,000.00\u003c/li\u003e\n\u003cli\u003echair price = Rp50,000.00\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eThe sum of two numbers is 28 and their difference is 12. The numbers are \u0026hellip;\nA. 8 and 20\u003cbr\u003e\nB. 10 and 18\u003cbr\u003e\nC. 12 and 16\u003cbr\u003e\nD. 14 and 14\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eAli is 30 years old now. Six years ago, Ali\u0026rsquo;s age was three times Budi\u0026rsquo;s age. Budi\u0026rsquo;s current age is \u0026hellip;\nA. 8 years\u003cbr\u003e\nB. 10 years\u003cbr\u003e\nC. 14 years\u003cbr\u003e\nD. 24 years\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eObserve the following graph.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbvf26NX9t0RmRitZg5epLmWDQyvPPIboa-Rxn7i27UV-Te1m7IofkgzbcYwlxZQJgBnA0FdDD-kGEtyT6th13KAIsoVFJbETxYiDRBoN09kxla_YUHUD8MbnFf5cJAnSlJ8qxH3vEoD4/s261/spldv2.png' alt=\"SLETV evaluation graph\" \u003e\u003c/p\u003e\n\u003cp\u003eThe equation that matches the graph is \u0026hellip;\nA. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x+3y=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nB. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x+y=9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nC. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x+y=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\nD. \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x+y=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003col start=\"4\"\u003e\n\u003cli\u003e\n\u003cp\u003eThe solution set of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x-2y=7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e14\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x+y=14\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e14\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\{(a,b)\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. The value of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea+b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u0026hellip;\nA. 9\u003cbr\u003e\nB. 7\u003cbr\u003e\nC. 5\u003cbr\u003e\nD. 4\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eOn a farm there are 13 animals consisting of chickens and goats. The total number of legs is 38. The number of goats is \u0026hellip;\nA. 5\u003cbr\u003e\nB. 6\u003cbr\u003e\nC. 7\u003cbr\u003e\nD. 8\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThat concludes this material on systems of linear equations in two variables.\u003c/p\u003e\n", "language": "en" }, { "title" : "Straight-Line Equations", "date_published" : "2020-10-19T16:09:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/persamaan-garis-lurus/", "url" : "https://www.aantriono.com/en/persamaan-garis-lurus/", "summary": " Introduction Basic Competencies3.4 Analyze linear functions as straight-line equations and interpret their graphs in contextual problems.\n4.4 Solve contextual problems related to linear functions as straight-line equations.\nLearning ObjectivesAfter studying this material, students are expected to be able to:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch3 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h3\u003e\u003cp\u003e3.4 Analyze linear functions as straight-line equations and interpret their graphs in contextual problems.\u003c/p\u003e\n\u003cp\u003e4.4 Solve contextual problems related to linear functions as straight-line equations.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtJ6zpwJDjFBzElp7v7nSafkXankguJzE6VuoTJ8NrnQzdiJzs1-EoccgC6D5JoLSt1mBNRZ4VQiMLZc-y4n8660qdp3e5H1JQ1nQwJrZj6kb1wrZhoWEjLRphjq77edkOcfVSSh-kD1deKEJap9-JsN9vsntd8faONA2WWP2ZO7TVbeY9x8dJd3lg/w640-h200/garis.png' alt=\"Straight-line equations\" \u003e\u003c/p\u003e\n\u003ch3 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h3\u003e\u003cp\u003eAfter studying this material, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eanalyze linear functions and their graphs,\u003c/li\u003e\n\u003cli\u003edetermine the slope of a line,\u003c/li\u003e\n\u003cli\u003edetermine the relationship between two lines from their slopes,\u003c/li\u003e\n\u003cli\u003eform a straight-line equation from the given information.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h2\u003e\u003ch3 id=\"1-graph-of-a-straight-line-equation\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-graph-of-a-straight-line-equation\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Graph of a Straight-Line Equation\"\u003e\u003c/a\u003e1. Graph of a Straight-Line Equation\u003c/h3\u003e\u003cp\u003eObserve the following graph.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgihYQifzbSDirK4k0gTNt7GmlTG960e9FSkjNFdgkNwT_aab3H87txbgUtoaM6UoN27wgE43CiTqcjpdRTgICwRRxL0-AwpOU11lx9H0ondWJez_4xrDfxRsuFwAuuOocRIDolSCOeLjM/s320/pgl1.png' alt=\"Line graph\" \u003e\u003c/p\u003e\n\u003cp\u003eFor the equations \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey = 2x\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey = 2x + 1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the coefficient of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is 2. This value describes the comparison between the change in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the change in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. It is called the \u003cstrong\u003eslope\u003c/strong\u003e or \u003cstrong\u003egradient\u003c/strong\u003e of the line.\u003c/p\u003e\n\u003cp\u003eIf the slope is denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then the general form of a straight-line equation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny = mx + c\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = slope,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e = constant.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eExamples of straight-line equations:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey = 3x - 4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2x - 3y - 5 = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eTo draw the graph of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey = mx + c\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003efind the intercept with the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis,\u003c/li\u003e\n\u003cli\u003efind the intercept with the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis,\u003c/li\u003e\n\u003cli\u003econnect the two points.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eDraw the graph of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny = -3x + 6\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFind the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-intercept by setting \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n0 \u0026amp;= -3x + 6 \\\\\n3x \u0026amp;= 6 \\\\\nx \u0026amp;= 2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-intercept is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eFind the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-intercept by setting \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex = 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\ny \u0026amp;= -3(0) + 6 \\\\\ny \u0026amp;= 6\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-intercept is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0,6)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZibymhCFV4mxYIfPFOWEy0jScGhgzTs0UuzFCcTJpMJ7nx836ykV6LJtFgZHtupr1ZuNhuTP18Jf4yA3yNIdTMfCQFDntWKEzOp2Cb9FyzetdGEmkb5K_f6w26QGthG9WzGizAhtqqxE/s320/gradien4.png' alt=\"Example graph\" \u003e\u003c/p\u003e\n\u003ch3 id=\"2-determining-the-slope-of-a-straight-line\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-determining-the-slope-of-a-straight-line\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Determining the Slope of a Straight Line\"\u003e\u003c/a\u003e2. Determining the Slope of a Straight Line\u003c/h3\u003e\u003cp\u003eThe slope shows the degree of inclination of a line relative to the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis and is usually denoted by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch4 id=\"a-slope-of-a-line-through-the-origin--and-a-point\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-slope-of-a-line-through-the-origin--and-a-point\" class=\"header-mark\" aria-label=\"Direct link to section: A. Slope of a line through the origin (0,0)(0,0)(0,0) and a point (x1,y1)(x_1,y_1)(x1​,y1​)\"\u003e\u003c/a\u003eA. Slope of a line through the origin \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and a point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_1,y_1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h4\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxFtsdh9RQAuHOILONrsc4MF6tnFjQbEJR3htzAB6cMFc_lNKuO71lVXg7hOqZAGpFhUcsfJAF6XZpDFmCXycqP4SA72Bzz53uBjh5LftPbpRUsLg1-UL8jVq8RcF1foFeGZ_D3LlVj-Y/s0/pgl3.png' alt=\"Line through the origin\" \u003e\u003c/p\u003e\n\u003cp\u003eFormula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm = \\frac{y_1}{x_1}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.9436em;vertical-align:-0.836em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.836em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"b-slope-of-a-line-crossing-the--axis-at--and-the--axis-at\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-slope-of-a-line-crossing-the--axis-at--and-the--axis-at\" class=\"header-mark\" aria-label=\"Direct link to section: B. Slope of a line crossing the xxx-axis at (b,0)(b,0)(b,0) and the yyy-axis at (0,a)(0,a)(0,a)\"\u003e\u003c/a\u003eB. Slope of a line crossing the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(b,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e-axis at \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0,a)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h4\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlgTWFaM4EOB3HcrQaBYSsFOSaWkzoVQ7u9HNsAie5M9vCnRKjfAJ6d508Y_zyDuEHuy8npn2HQXw3Mc-heI8Awf4wu4GwcZ5_8gnjASTZNeK_HfxNJoXcRf4SltHDjI0_ADbP6GHgDo0/s0/gradien1.png' alt=\"Intercept form line\" \u003e\u003c/p\u003e\n\u003cp\u003eFormula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm = \\frac{-a}{b}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.9463em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"c-slope-of-a-line-through-two-points--and\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-slope-of-a-line-through-two-points--and\" class=\"header-mark\" aria-label=\"Direct link to section: C. Slope of a line through two points (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​)\"\u003e\u003c/a\u003eC. Slope of a line through two points \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_1,y_1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_2,y_2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h4\u003e\u003cp\u003eFormula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm = \\frac{y_2 - y_1}{x_2 - x_1}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0963em;vertical-align:-0.836em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.836em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eFind the slope of the line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA(-1,3)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB(7,-5)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nm_{AB} \u0026amp;= \\frac{-5 - 3}{7 - (-1)} \\\\\n \u0026amp;= \\frac{-8}{8} \\\\\n \u0026amp;= -1\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6.3649em;vertical-align:-2.9324em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.4324em;\"\u003e\u003cspan style=\"top:-5.4324em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3283em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eA\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.875em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.049em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9324em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.4324em;\"\u003e\u003cspan style=\"top:-5.4324em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.875em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.049em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.9324em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"slope-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#slope-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: Slope Simulation\"\u003e\u003c/a\u003eSlope Simulation\u003c/h3\u003e\u003cp\u003eUse the following simulation to better understand slope visually.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/y5n3r8pt/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"gradien\" width=\"700px\"\u003e\u003c/iframe\u003e\n\u003ch3 id=\"3-relationship-between-two-lines-from-their-slopes\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-relationship-between-two-lines-from-their-slopes\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Relationship Between Two Lines from Their Slopes\"\u003e\u003c/a\u003e3. Relationship Between Two Lines from Their Slopes\u003c/h3\u003e\u003ch4 id=\"a-parallel-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-parallel-lines\" class=\"header-mark\" aria-label=\"Direct link to section: A. Parallel lines\"\u003e\u003c/a\u003eA. Parallel lines\u003c/h4\u003e\u003cp\u003eTwo parallel lines have the same slope.\u003c/p\u003e\n\u003cp\u003eExample:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eThe line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(1,6)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(3,2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eThe line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2,-9)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-3,1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8ToRXtx4kNuKW-k5a6SLPA4Xrxp54t5C28rkNv5aQrthT5CY6ljIKXsSL0LNcD254YqBqxPJihvGJ0sSquutgmU4qeNpbWibf0dRSiCjFm_FN_U0PFKhS7o4PJ61LvH_Tph4qNR6O8RY/s320/gradien5.png' alt=\"Parallel lines\" \u003e\u003c/p\u003e\n\u003cp\u003eFirst slope:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm_1 = \\frac{2-6}{3-1} = \\frac{-4}{2} = -2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSecond slope:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm_2 = \\frac{1-(-9)}{-3-2} = \\frac{10}{-5} = -2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1963em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.427em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSince \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em_1 = m_2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the lines are parallel.\u003c/p\u003e\n\u003ch4 id=\"b-perpendicular-lines\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-perpendicular-lines\" class=\"header-mark\" aria-label=\"Direct link to section: B. Perpendicular lines\"\u003e\u003c/a\u003eB. Perpendicular lines\u003c/h4\u003e\u003cp\u003eTwo lines are perpendicular if:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm_1 \\times m_2 = -1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7333em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eExample:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eThe line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0,-8)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(3,1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eThe line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-3,3)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(6,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3Cv51171SedFwCKXacP_bdklq0TjM7xUUJ4TGvQeceexP5Qq4wKdd9bwukJMqLPQT_kym7v-1obj_D3ax0Vkq72VRZW4rZsUgqiQtFM3RWD9dZM3QzE1agUXQWnQlHjqM16m3fQjMq_4/s320/gradien6.png' alt=\"Perpendicular lines\" \u003e\u003c/p\u003e\n\u003cp\u003eFirst slope:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm_1 = \\frac{1-(-8)}{3-0} = \\frac{9}{3} = 3\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1963em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.427em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSecond slope:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nm_2 = \\frac{0-3}{6-(-3)} = \\frac{-3}{9} = -\\frac{1}{3}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2574em;vertical-align:-0.936em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eBecause:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3 \\times \\left(-\\frac{1}{3}\\right) = -1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ethe lines are perpendicular.\u003c/p\u003e\n\u003ch3 id=\"4-forming-a-straight-line-equation\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-forming-a-straight-line-equation\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Forming a Straight-Line Equation\"\u003e\u003c/a\u003e4. Forming a Straight-Line Equation\u003c/h3\u003e\u003ch4 id=\"a-through-one-point-with-slope\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-through-one-point-with-slope\" class=\"header-mark\" aria-label=\"Direct link to section: A. Through one point with slope mmm\"\u003e\u003c/a\u003eA. Through one point with slope \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/h4\u003e\u003cp\u003eIf a line passes through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_1,y_1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and has slope \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny - y_1 = m(x - x_1)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eFind the equation of a line with slope 3 passing through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-2,5)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\ny - 5 \u0026amp;= 3(x - (-2)) \\\\\ny - 5 \u0026amp;= 3(x + 2) \\\\\ny - 5 \u0026amp;= 3x + 6 \\\\\ny \u0026amp;= 3x + 11\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:6em;vertical-align:-2.75em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.25em;\"\u003e\u003cspan style=\"top:-5.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e))\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.75em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the equation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny = 3x + 11\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"b-through-two-points\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-through-two-points\" class=\"header-mark\" aria-label=\"Direct link to section: B. Through two points\"\u003e\u003c/a\u003eB. Through two points\u003c/h4\u003e\u003cp\u003eIf a line passes through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_1,y_1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x_2,y_2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{y-y_1}{y_2-y_1} = \\frac{x-x_1}{x_2-x_1}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1408em;vertical-align:-0.8804em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8804em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0963em;vertical-align:-0.836em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.836em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny-y_1 = \\frac{y_2-y_1}{x_2-x_1}(x-x_1)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0963em;vertical-align:-0.836em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.2603em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan 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style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.836em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eFind the equation of the line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(10,3)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(2,7)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\frac{y-3}{7-3} \u0026amp;= \\frac{x-10}{2-10} \\\\\n\\frac{y-3}{4} \u0026amp;= \\frac{x-10}{-8} \\\\\ny-3 \u0026amp;= -\\frac{1}{2}(x-10) \\\\\ny-3 \u0026amp;= -\\frac{1}{2}x + 5 \\\\\ny \u0026amp;= -\\frac{1}{2}x + 8\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:11.7039em;vertical-align:-5.6019em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.1019em;\"\u003e\u003cspan style=\"top:-8.1019em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.7112em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3204em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.013em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.2945em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.6019em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.1019em;\"\u003e\u003cspan style=\"top:-8.1019em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.7112em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3204em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.013em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.2945em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:5.6019em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo the equation is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny = -\\frac{1}{2}x + 8\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eAnswer the following questions to test your understanding.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eFind the slope of the line in the figure below.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmuDYp3KMLT8QGcihpKi3VpCh10G6H1Xmyl3CGBn5yMZGIWOvXMfrZ8BXw8qxdNTjwk-ZRgey40vlrYHci-dpGMDyP8PLziFe2TCn07bSj94caCXviSu-zzIvlZIW0bSI8yY4db8aMrTc/s320/gradien7.png' alt=\"Slope question\" \u003e\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eThe line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP(2,y)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ(5,7)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e has slope \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e-1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Find \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eLine \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eg\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eg\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eg\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is perpendicular to the line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP(0,7)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eQ\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eQ(2,1)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eQ\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Find the slope of line \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eg\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eg\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003eg\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eFind the equation of the line through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(1,2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and perpendicular to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3x+4y=8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eFind the equation of a line with slope \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\frac{2}{3}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e passing through \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-6,-5)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n", "language": "en" }, { "title" : "Dilation", "date_published" : "2020-10-06T14:53:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/dilatasi/", "url" : "https://www.aantriono.com/en/dilatasi/", "summary": " Introduction\nLearning ObjectivesAfter studying dilation, students are expected to be able to:\nExplain dilation as a geometric transformation in contextual problems. Solve contextual problems involving dilation. MaterialDigital cameras often provide a zoom feature that enlarges or reduces an image. This idea is closely related to dilation in geometry.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD5JIBHsJEjIA71iMHmpUZBln5HStrLDgU0FGqGw3_1gcRiGLAH08cWUfUYbsmQLHRphCW-4Xa42Ea-ISXswJ7MCnur_y71MeMZmi7bP85GmdfYAvXO-oC7BFjtjIGBQJE9AutxTCHR8k/s604/judul+dilatasi.png' alt=\"Dilation illustration\" \u003e\u003c/p\u003e\n\u003ch3 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h3\u003e\u003cp\u003eAfter studying dilation, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eExplain dilation as a geometric transformation in contextual problems.\u003c/li\u003e\n\u003cli\u003eSolve contextual problems involving dilation.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h2\u003e\u003cp\u003eDigital cameras often provide a \u003cem\u003ezoom\u003c/em\u003e feature that enlarges or reduces an image. This idea is closely related to dilation in geometry.\u003c/p\u003e\n\u003cp\u003eDilation is a transformation that changes the size of a figure, either enlarging or reducing it, without changing its shape.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgziSnZHe4p2AxauU-HmjPYvuHpXFw0eblDSO_AmSMYcI_E0KUIMoTnV3bs43m_cUCmUMoMlT6_GUxxMPHmi5tdfzBG8t2hkVwf7pEEteOyZ-36P85f4oABZ-IC8efNmTImkk3FVAKQP4o/s476/dilatasi+1.png' alt=\"Example of dilation\" \u003e\u003c/p\u003e\n\u003cp\u003eIf A is the original figure before dilation, then B, C, and D are images obtained from dilation with certain scale factors.\u003c/p\u003e\n\u003cp\u003eTwo pieces of information are needed to perform a dilation:\u003c/p\u003e\n\u003ch3 id=\"scale-factor\" class=\"headerLink\"\u003e\n \u003ca href=\"#scale-factor\" class=\"header-mark\" aria-label=\"Direct link to section: Scale Factor\"\u003e\u003c/a\u003eScale Factor\u003c/h3\u003e\u003cp\u003eThe scale factor determines the amount of enlargement or reduction. Its value may be positive, zero, or negative.\u003c/p\u003e\n\u003ch3 id=\"center-of-dilation\" class=\"headerLink\"\u003e\n \u003ca href=\"#center-of-dilation\" class=\"header-mark\" aria-label=\"Direct link to section: Center of Dilation\"\u003e\u003c/a\u003eCenter of Dilation\u003c/h3\u003e\u003cp\u003eThe center of dilation is a coordinate point. If it is not specified, the center is usually taken as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eO(0,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eTriangle ABC has vertices A(1, 3), B(2, 3), and C(2, 1). Determine the image of the triangle after a dilation with scale factor 3 and center at the origin.\u003c/p\u003e\n\u003ch3 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eThe steps are as follows.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ePlot points A, B, and C on the Cartesian plane.\u003c/li\u003e\n\u003cli\u003eMultiply each coordinate by the scale factor 3:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo 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separator=\"true\"\u003e,\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi mathvariant=\"normal\"\u003e.\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nA\u0026#x27; \u0026amp;= (3 \\times 1,\\ 3 \\times 3) = (3,9),\\\\\nB\u0026#x27; \u0026amp;= (3 \\times 2,\\ 3 \\times 3) = (6,9),\\\\\nC\u0026#x27; \u0026amp;= (3 \\times 2,\\ 3 \\times 1) = (6,3).\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5em;vertical-align:-2em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5em;\"\u003e\u003cspan style=\"top:-4.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.16em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.66em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mord\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the image of triangle ABC is triangle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;B\u0026#x27;C\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e with vertices \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;(3,9)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u0026#x27;(6,9)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u0026#x27;(6,3)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhICyWXQOMGonVS-7yM2U9ER41V9AuKSghOc_P8_grtlmzwe_BLrPqzeADjj02fwGj0-RSkXKVY4kpVaCOYPXp4JHpMZjiUSLgTB3YxkSr30fOPeHpmNgEasfMcJ1eqlsPMBoQBRXIT_EU/s547/contoh+dilatasi+1.png' alt=\"First dilation example\" \u003e\u003c/p\u003e\n\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eRectangle ABCD has vertices A(2, 3), B(5, 3), C(5, 1), and D(2, 1). Determine the coordinates of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;B\u0026#x27;C\u0026#x27;D\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the image of rectangle ABCD after a dilation with center P(1, 4) and scale factor 3.\u003c/p\u003e\n\u003ch3 id=\"solution-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-1\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003col\u003e\n\u003cli\u003ePlot the center of dilation \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eP(1,4)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.13889em;\"\u003eP\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the rectangle ABCD on the coordinate plane.\u003c/li\u003e\n\u003cli\u003eDraw rays from point P through each vertex so that each image point follows the given scale factor.\u003c/li\u003e\n\u003cli\u003eThe image points are:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmsup\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA\u0026#x27;(4,1),\\quad B\u0026#x27;(13,1),\\quad C\u0026#x27;(13,-5),\\quad D\u0026#x27;(4,-5)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0519em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"4\"\u003e\n\u003cli\u003eConnect points \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eD\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to form rectangle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;B\u0026#x27;C\u0026#x27;D\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgibtmK16JcpGFDgX7xZK20MNrih8cAODIKuGfqDvuOulQwc3jyg8aQjYN4tpd72jPWT72dk7YmhiZEt33gozyKkcAaT9uN2x9BzR2KRAgQYZB9znjCHmn6DvWaFfLcUh4qfSF87_29io4/s786/dilatasi++contoh+2.png' alt=\"Second dilation example\" \u003e\u003c/p\u003e\n\u003ch3 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h3\u003e\u003cp\u003eThe image of point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x,y)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e after dilation with scale factor \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and center \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eO\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eO(0,0)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eO\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ek\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(kx,\\ ky)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"simulations\" class=\"headerLink\"\u003e\n \u003ca href=\"#simulations\" class=\"header-mark\" aria-label=\"Direct link to section: Simulations\"\u003e\u003c/a\u003eSimulations\u003c/h2\u003e\u003ch3 id=\"1-slider-based-image-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-slider-based-image-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Slider-based image simulation\"\u003e\u003c/a\u003e1. Slider-based image simulation\u003c/h3\u003e\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/nznqcc5p/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Dilatasiku\" width=\"700px\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"2-triangle-abc-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-triangle-abc-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Triangle ABC simulation\"\u003e\u003c/a\u003e2. Triangle ABC simulation\u003c/h3\u003e\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/hxha7ht6/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Dilatasi 2\" width=\"700px\"\u003e \u003c/iframe\u003e\n\u003ch3 id=\"3-uzumaki-naruto-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-uzumaki-naruto-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Uzumaki Naruto simulation\"\u003e\u003c/a\u003e3. Uzumaki Naruto simulation\u003c/h3\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwbuGlpQTwmZFcn3SxGgiJVJ0qIsfb3MmHBfy-9PzKdYeQg5JxBYb33Uezf-8x9aJLL_6pp6Acex28CH_aZCHDCbC4w_oFmd8LgeJ1RtpKVlCb1wOd4YHVPPlqxCrwXi0c302k4ZD9_KA/s1150/naruto+dilatasi.gif' alt=\"Naruto dilation animation\" \u003e\u003c/p\u003e\n\u003ch3 id=\"4-two-circle-dilation-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-two-circle-dilation-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Two-circle dilation simulation\"\u003e\u003c/a\u003e4. Two-circle dilation simulation\u003c/h3\u003e\u003cp\u003eUse a laptop for the best result.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/kgmnxwvf/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"simulasi 2 lingkaran\" width=\"700px\"\u003e \u003c/iframe\u003e\n\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eAfter studying this topic, solve the following problems.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ePoint A(2, 3) is dilated with center O(0, 0) and scale factor 2. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;(x,y)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the image point, determine the coordinates of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/li\u003e\n\u003cli\u003eLook at the following figure.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqdCFomzNd6ro4LLjLkYb0v2VuhpYrNYkA5PB5rGNE3RenKmKVFUG3nspj1GB9hy_lCejPqtBJL8KcuUe5VPu5eWsqYuoB3H0kbfc5cwIumHuoxGuktvTzk_fVl1OqZhyphenhyphenqaIbbyuE1Big/s757/dilatasi+soal.png' alt=\"Dilation exercise\" \u003e\u003c/p\u003e\n\u003cp\u003eParallelogram \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;B\u0026#x27;C\u0026#x27;D\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the image of parallelogram ABCD under a dilation with center P and scale factor \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ek\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ek\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03148em;\"\u003ek\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. From the picture, determine:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ethe coordinates of the center point P;\u003c/li\u003e\n\u003cli\u003ethe scale factor of the dilation.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat is the discussion of dilation as part of geometric transformations. I hope this material is useful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Rotation", "date_published" : "2020-10-05T14:20:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/uji-coba/", "url" : "https://www.aantriono.com/en/uji-coba/", "summary": " Introduction\nA. Learning ObjectivesAfter studying the rotation material (rotation) This is expected to:\nExplain geometric transformations (rotation) which is connected to contextual problems.\nSolve contextual problems related to geometric transformations (rotation).\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrj645TaeMCPUoY4k55O04HLoyvZVSXVwPBTmTYfqlO6rOdHqf5xdhIvwGctqou4Bxgtl5F-v2pFiwiOt13KCo50jhLPJZB82-sBVc68JRIAl7lbvipMXGeCQdIs6lEglk99Yymc2J1xg/s604/ROTASI\n1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"575\" data-original-width=\"604\" height=\"304\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrj645TaeMCPUoY4k55O04HLoyvZVSXVwPBTmTYfqlO6rOdHqf5xdhIvwGctqou4Bxgtl5F-v2pFiwiOt13KCo50jhLPJZB82-sBVc68JRIAl7lbvipMXGeCQdIs6lEglk99Yymc2J1xg/w320-h304/ROTASI\n1.png\" title=\"Gambar rotasi suatu bangun\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"a-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: A. Learning Objectives\"\u003e\u003c/a\u003eA. Learning Objectives\u003c/h2\u003e\u003cp\u003eAfter studying the rotation material (rotation) This is expected to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eExplain geometric transformations (rotation) which is connected to contextual problems.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSolve contextual problems related to geometric transformations (rotation).\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"b-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-material\" class=\"header-mark\" aria-label=\"Direct link to section: B. Material\"\u003e\u003c/a\u003eB. Material\u003c/h2\u003e\u003cp\u003eHave you ever ridden a merry-go-round? When you ride the merry-go-round, you will see that the carousel revolves around a pole in the middle. While rotating the pole remains in place. In the case of a merry-go-round, the pole functions as the center point of rotation, while those who are rotated are the people sitting on the merry-go-round. Rotation is a form of transformation that rotates each point in the image to a certain angle and direction towards a fixed point. This fixed point is called the center of rotation. The angle of the object\u0026rsquo;s image with respect to its initial position is called the rotation angle. Look at the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhl42eiL2YPNOGT3iOMKy9lusSJm1VzoWRYQ1_JxxuWS20YyZSQWleXGtlw2CXIIj-47Ytdbjji66OJwnYzxW3PskvNGr2Bdu8qbkG-5EjV-D9PcvDmEV0aoTfLTGQsZrfLIpXZNKQROdc/s291/rotasi2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"241\" data-original-width=\"291\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhl42eiL2YPNOGT3iOMKy9lusSJm1VzoWRYQ1_JxxuWS20YyZSQWleXGtlw2CXIIj-47Ytdbjji66OJwnYzxW3PskvNGr2Bdu8qbkG-5EjV-D9PcvDmEV0aoTfLTGQsZrfLIpXZNKQROdc/s0/rotasi2.png\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eTo be able to rotate, information is needed about the angle of rotation, direction of rotation and center point of rotation.* Rotation angleThe rotation angle is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e0^0-360^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e* Direction of rotationBased on the rules for determining quadrants, the direction of rotation is positive if counterclockwise and the direction of rotation is negative if clockwise.* Center point of rotationThe center of rotation is a coordinate point. If the center of rotation is not specified then the center of rotation is the origin \u003cem\u003eO\u003c/em\u003e(0.0)**Example of the first question:**A point \u003cem\u003eA\u003c/em\u003e(2.4) rotated with the center of rotation being the origin \u003cem\u003eO\u003c/em\u003e(0.0). Determine the rotation result of point A if it is rotated counterclockwise by:a. rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003eb. rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003ec. rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e270\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e270^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e27\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003ed. rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e**Solution:**The steps to get the rotation results from point A are as follows:(i) Measure the distance from the point (0, 0) to the point \u003cem\u003eA\u003c/em\u003e(2, 4)(ii) Rotate it with the center point (0, 0) as far as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e counterclockwise.(iii) The point obtained is the result of rotation \u003cem\u003eA\u003c/em\u003e(2, 4), that is \u003cem\u003eA\u0026rsquo;\u003c/em\u003e(-4, 2).(iv) Repeat steps (i) - (iii) for rotation angles \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e270\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e270^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e27\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.a. Point rotation results \u003cem\u003eA\u003c/em\u003e(2, 4) for rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cem\u003eA\u0026rsquo;\u003c/em\u003e(-4, 2)b. Point rotation results \u003cem\u003eA\u003c/em\u003e(2, 4) for rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e180\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e180^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e18\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cem\u003eA\u0026rsquo;\u0026rsquo;\u003c/em\u003e(-2, -4)a. Point rotation results \u003cem\u003eA\u003c/em\u003e(2, 4) for rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e270\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e270^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e27\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cem\u003eA\u0026rsquo;\u0026rsquo;\u0026rsquo;\u003c/em\u003e(4, -2)a. Point rotation results \u003cem\u003eA\u003c/em\u003e(2, 4) for rotation angle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cem\u003eA\u0026rsquo;\u0026rsquo;\u0026rsquo;\u0026rsquo;\u003c/em\u003e(2, 4)To make it clearer, see the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhES2QqbOsew5EURo9Votjp1zNdyPg8plrk_LH0TpD-g0HsOZAJ-Y5zB5DRIw2qhoO4WaTtC_6Azh1o8W8jmXKOQobDz0wJP8MOVNvLmxDKR27333iWUYKjmyUkcd7SZbge6nQR_rWOua0/s556/rotasi7.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"512\" data-original-width=\"556\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhES2QqbOsew5EURo9Votjp1zNdyPg8plrk_LH0TpD-g0HsOZAJ-Y5zB5DRIw2qhoO4WaTtC_6Azh1o8W8jmXKOQobDz0wJP8MOVNvLmxDKR27333iWUYKjmyUkcd7SZbge6nQR_rWOua0/s320/rotasi7.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e**Example of the second question:**A triangle ABC with \u003cem\u003eA\u003c/em\u003e(1, 4), \u003cem\u003eB\u003c/em\u003e(1, 6), And \u003cem\u003eC\u003c/em\u003e(3, 6) rotated with a center point \u003cem\u003eO\u003c/em\u003e(0, 0) of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e clockwise. Describe the results of the rotation.**Solution:**To get the rotation results of triangle ABC, do the following steps:(i) Measure the distance from the point (0, 0) to the point \u003cem\u003eA\u003c/em\u003e(1, 4)(ii) Rotate it with the center (0, 0) as far as \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e90\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e90^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e clockwise.(iii) The point obtained is the result of rotation \u003cem\u003eA\u003c/em\u003e(1, 4), that is \u003cem\u003eA\u0026rsquo;\u003c/em\u003e(4, -1).(iv) Do the same activities for point B and point C to obtain \u003cem\u003eB\u003c/em\u003e(6, -1) And \u003cem\u003eC\u003c/em\u003e(6, -3)(v) Connect points A\u0026rsquo;, B\u0026rsquo;, and point C\u0026rsquo; to obtain a triangle A\u0026rsquo;B\u0026rsquo;C\u0026rsquo; as shown in the following image.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmeEglbOg_o3W7wptAAUVlnQQdujC31hynvDfrh5ruV1RYR8gjYedbOVF39g64ftzp3OTg4jmkx6oNiML6B11hi6HJZs-WOrfgB-xsKUKdRGffhDBaP0NrTSo8WJjbEHb6oiChwRmja7Q/s549/rotasi8.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"549\" data-original-width=\"456\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmeEglbOg_o3W7wptAAUVlnQQdujC31hynvDfrh5ruV1RYR8gjYedbOVF39g64ftzp3OTg4jmkx6oNiML6B11hi6HJZs-WOrfgB-xsKUKdRGffhDBaP0NrTSo8WJjbEHb6oiChwRmja7Q/s320/rotasi8.png\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eRotational Symmetry\u003c/strong\u003eRotational symmetry is closely related to rotation. A shape has rotational symmetry if it can be rotated at its center and returned to its original position more than once in one complete rotation, as in the picture below.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9BVLBZFfTRNrlYqpbU1Z2DiIVRFeCCzjsXTKGnKwMf6ia_ZHg8JnSx0a-oZEK-RMHffCwZQcwkUhxtpxfIx_naPBQ3aDyGShcnA3epA753EFZ-l6cy1PSprhGh3SryjD0UOylHMgFGts/s724/simetri1.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"362\" data-original-width=\"724\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9BVLBZFfTRNrlYqpbU1Z2DiIVRFeCCzjsXTKGnKwMf6ia_ZHg8JnSx0a-oZEK-RMHffCwZQcwkUhxtpxfIx_naPBQ3aDyGShcnA3epA753EFZ-l6cy1PSprhGh3SryjD0UOylHMgFGts/s320/simetri1.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThe number of times it returns to its original position in one complete rotation or \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called the degree of rotational symmetry. All figures will return to their original position at least once when rotated \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e360\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e360^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e36\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. So every shape has a degree of rotational symmetry of at least one. Only shapes that have a degree of rotational symmetry of 2 or more are said to have rotational symmetry.**Conclusion :**From the rotation results in the example problem, it can be concluded that the rotation results of a point with a center of rotation O(0, 0) and rotated counterclockwise as shown in the following table:\u003c/p\u003e\n\u003cstyle type=\"text/css\"\u003e\n.tg {border-collapse:collapse;border-color:#aaa;border-spacing:0;}\n.tg td{background-color:#fff;border-color:#aaa;border-style:solid;border-width:1px;color:#333;\n font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg th{background-color:#f38630;border-color:#aaa;border-style:solid;border-width:1px;color:#fff;\n font-family:Arial, sans-serif;font-size:14px;font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg .tg-0lax{text-align:left;vertical-align:top}\n.tg .tg-dg7a{background-color:#FCFBE3;text-align:left;vertical-align:top}\n\u003c/style\u003e\n\u003ctable class=\"tg\"\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n\u003cth class=\"tg-0lax\"\u003eRotation Angle\u003c/th\u003e\n\u003cth class=\"tg-0lax\"\u003eStarting Point\u003c/th\u003e\n\u003cth class=\"tg-0lax\"\u003eShadow\u003c/th\u003e\n\u003cth class=\"tg-0lax\"\u003eExample\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-dg7a\"\u003e$90^0$\u003c/td\u003e\n\u003ctd class=\"tg-dg7a\"\u003e*A(x,y)*\u003c/td\u003e\n\u003ctd class=\"tg-dg7a\"\u003e*A(-y, x)*\u003c/td\u003e\n\u003ctd class=\"tg-dg7a\"\u003e*A*(2, 4) become *A'*(-4, 2)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-0lax\"\u003e$180^0$\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003e*A(x,y)*\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003e*A(-x -,y)*\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003e*A*(2, 4) become *A'*(-2, -4)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-dg7a\"\u003e$270^0$\u003c/td\u003e\n\u003ctd class=\"tg-dg7a\"\u003e*A(x,y)*\u003c/td\u003e\n\u003ctd class=\"tg-dg7a\"\u003e*A(y, -x)*\u003c/td\u003e\n\u003ctd class=\"tg-dg7a\"\u003e*A*(2, 4) become *A'*(4, -2)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-0lax\"\u003e$360^0$\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003e*A(x,y)*\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003e*A(x,y)*\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003e*A*(2, 4) become *A'*(2, 4)\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e**Simulation:**Below we present a rotation simulation of a quadrilateral ABCD. 1. Move the quadrilateral ABCD or writing corners to the right and to the left.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/byvw9p5k/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Rotasi\" width=\"700px\"\u003e \u003c/iframe\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eBelow we present a rotation simulation of a triangle ABC.Slide/Move point D or triangle ABC in any direction\u003c/li\u003e\n\u003c/ol\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/ew8qqucv/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Simulasi rotasi segitiga\" width=\"700px\"\u003e \u003c/iframe\u003e\n\u003ch2 id=\"c-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: C. Evaluation\"\u003e\u003c/a\u003eC. Evaluation\u003c/h2\u003e\u003cp\u003eAfter studying this rotation material, please do the following questions:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eThe flat shape A\u0026rsquo;B\u0026rsquo;C\u0026rsquo;D\u0026rsquo;E\u0026rsquo;F\u0026rsquo; below is an image of the flat shape ABCDEF after undergoing rotation with the center of rotation \u003cem\u003eO\u003c/em\u003e(0, 0) and a certain rotation angle.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgx3nbmquiqZXtX2868RWHnHBkzkG0YFBenqeQ0dDuPGf_agTwSSVxsi-nw4aeXS3S-HiiuIK-d5UVpPTpAGH_tt8-HNdQHqdXVLbfHs0f9dpF_XJwp1d5iWv9vM5xQG2UxlCSNZRsFlRI/s708/rotasi9.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"521\" data-original-width=\"708\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgx3nbmquiqZXtX2868RWHnHBkzkG0YFBenqeQ0dDuPGf_agTwSSVxsi-nw4aeXS3S-HiiuIK-d5UVpPTpAGH_tt8-HNdQHqdXVLbfHs0f9dpF_XJwp1d5iWv9vM5xQG2UxlCSNZRsFlRI/s320/rotasi9.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003ea. Determine the angle of rotation.b. Is the direction of rotation counterclockwise?\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003ePoint \u003cem\u003eB\u003c/em\u003e(2, 1) rotated with a rotation angle of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e270\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e270^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e27\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e counterclockwise. Determine the result of the rotation. This is the material for rotational submaterial geometric transformation. Hopefully it\u0026rsquo;s useful, Thank you. 😊🙏\u003c/li\u003e\n\u003c/ol\u003e\n", "language": "en" }, { "title" : "Translation", "date_published" : "2020-10-01T22:43:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/translasi/", "url" : "https://www.aantriono.com/en/translasi/", "summary": " Introduction Learning ObjectivesAfter studying translation, students are expected to be able to:\nExplain translation as a geometric transformation in contextual problems. Solve contextual problems involving translation. MaterialWhat is translation? What happens to the size and shape of an object after it is shifted?\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch3 id=\"learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Objectives\"\u003e\u003c/a\u003eLearning Objectives\u003c/h3\u003e\u003cp\u003eAfter studying translation, students are expected to be able to:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eExplain translation as a geometric transformation in contextual problems.\u003c/li\u003e\n\u003cli\u003eSolve contextual problems involving translation.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYjKSXdF7oJXrDypGsXUmhXKHR9LbAZ5bwsmZQm2Ig8rKTyv9OdjK07ciC0oLCkBoBFg8sIFfJ0_XfRxOZC0EHb6j1lpuO7oVawi5WlKH3wW3FtRkb05TGGjGlIFBdxBMm21X332zjWvenw6zaDw3eWUlJnoBtkKYrJ43m_7X3ph3u-ZLN7oK-1Yf8/s1280/www.aantriono.com.png' alt=\"Translation illustration\" \u003e\u003c/p\u003e\n\u003ch2 id=\"material\" class=\"headerLink\"\u003e\n \u003ca href=\"#material\" class=\"header-mark\" aria-label=\"Direct link to section: Material\"\u003e\u003c/a\u003eMaterial\u003c/h2\u003e\u003cp\u003eWhat is translation? What happens to the size and shape of an object after it is shifted?\u003c/p\u003e\n\u003cp\u003eTranslation is a geometric transformation that moves a figure from one position to another without changing its size, shape, or orientation. A figure may be shifted upward, downward, to the left, or to the right.\u003c/p\u003e\n\u003cp\u003eLook at the following illustration.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKJ9i6kXt2CVAjdBsUt7qItcrI1VLNr55cwTrOO71kHw0ftjwSzY6U2j418vnx6U0zsnqmOpjfF2CwS2n5dbfmyH7r0aBdZPdjKnh70n59PKwrBdKVpsN7bXhZhH3Gcjzd_CJVIFuK1Is/s599/Screenshot_1.png' alt=\"Example of translation\" \u003e\u003c/p\u003e\n\u003cp\u003eFrom the picture above, we can conclude that when a figure is translated:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eeach point moves in the same direction and by the same distance;\u003c/li\u003e\n\u003cli\u003ethe figure keeps the same orientation, size, and shape as before the translation.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eTo perform a translation, we need to know how far a point moves horizontally and vertically. This displacement is represented by the translation vector\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\binom{a}{b}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.1076em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e represents the horizontal shift;\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e represents the vertical shift.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is positive, the figure moves to the right. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is negative, it moves to the left. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is positive, it moves upward. If \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is negative, it moves downward.\u003c/p\u003e\n\u003ch3 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003ePoint \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA(2,4)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is translated by the vector \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{3}{-2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2984em;vertical-align:-0.4033em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8951em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.4033em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Determine the image of point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003ePoint \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e moves three units to the right and two units downward, so:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA\u0026#x27;(2+3,\\ 4-2)=A\u0026#x27;(5,2)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0519em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0519em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8019em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the image of point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;(5,2)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0019em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2wYr9lRVORNVbAYmSrXS2koAfnbTI8q0XYMqLqSfDiZF5PHsM8h8Ekyn-Sd-d3IczQY1XgRfkqzHT3jCPKtuDPc4Dk5_JwjXF8iylTVRgn1uzGXcGQxAxIbti9CqvDx3xZ2gZ4Qh7v60/s465/Screenshot_3.png' alt=\"Translated point\" \u003e\u003c/p\u003e\n\u003ch3 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h3\u003e\u003cp\u003eIf a point \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(x,y)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is translated by \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac linethickness=\"0px\"\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\binom{a}{b}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7454em;\"\u003e\u003cspan style=\"top:-2.355em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.144em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then its image is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmtext\u003e \u003c/mtext\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(x+a,\\ y+b)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\"\u003e \u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: Evaluation\"\u003e\u003c/a\u003eEvaluation\u003c/h2\u003e\u003cp\u003eAfter studying this topic, solve the following problems.\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ePoint \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eC(-1,3)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is translated 3 units to the right and 4 units downward. Determine its image.\u003c/li\u003e\n\u003cli\u003eLook at the following figure.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1CuQFSO9gZDX72yeh7Lo0yRo8_UOzfIwZkgWHf0EPfuODIDdwuzpEKe-oRnA6EsOO91YdeRtojnJ9Cm9CRJ4RJaFe-zRpMxYWk4lg6-JNuVYY-AXW3aSuFv0VB2pC39JF6JdVSGWQN6c/s551/Screenshot_4.png' alt=\"Translation exercise\" \u003e\u003c/p\u003e\n\u003cp\u003eDetermine the translation required to obtain rectangle \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eC\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\"\u003e′\u003c/mo\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u0026#x27;B\u0026#x27;C\u0026#x27;D\u0026#x27;\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7519em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.07153em;\"\u003eC\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.02778em;\"\u003eD\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7519em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e′\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eThat is the discussion of translation as part of geometric transformations. I hope this material is useful.\u003c/p\u003e\n", "language": "en" }, { "title" : "Geometric Transformation: Reflection", "date_published" : "2020-09-29T15:57:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/geometri-transformasi/", "url" : "https://www.aantriono.com/en/geometri-transformasi/", "summary": "\n**\nA. Basic Competencies 3.5 Explain geometric transformations (reflection, translation, rotation and dilation) which is connected to contextual problems.4.5 Resolve contextual problems related to geometric transformation (reflection, translation, rotation and dilation) B. Learning Objectives After studying the Geometry Transformation material, you are expected to be able to: Explain geometric transformations (reflection, translation, rotation and dilation) which is connected to contextual problems.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioPxW2oya1c8kFlNBI7Pt_Z0axYo7s0SGCPMISWZHZnE6r4UqPLrjw3VWtucsRQSPG-_xC5hpqK_FdjThXdM3zqIMprp5Orm_aaRdzCgKdL_ISckkhqbPc4F6Kma-YbcsJD6e_3eKu204/s515/refleksi1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"403\" data-original-width=\"515\" height=\"250\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioPxW2oya1c8kFlNBI7Pt_Z0axYo7s0SGCPMISWZHZnE6r4UqPLrjw3VWtucsRQSPG-_xC5hpqK_FdjThXdM3zqIMprp5Orm_aaRdzCgKdL_ISckkhqbPc4F6Kma-YbcsJD6e_3eKu204/w320-h250/refleksi1.png\" title=\"Gambar refleksi bangun datar segitiga ABC\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e**\u003c/p\u003e\n\u003ch2 id=\"a-basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: A. Basic Competencies\"\u003e\u003c/a\u003eA. Basic Competencies\u003c/h2\u003e\u003ch2 id=\"35-explain-geometric-transformations-reflection-translation-rotation-and-dilation-which-is-connected-to-contextual-problems45-resolve-contextual-problems-related-to-geometric-transformation-reflection-translation-rotation-and-dilation\" class=\"headerLink\"\u003e\n \u003ca href=\"#35-explain-geometric-transformations-reflection-translation-rotation-and-dilation-which-is-connected-to-contextual-problems45-resolve-contextual-problems-related-to-geometric-transformation-reflection-translation-rotation-and-dilation\" class=\"header-mark\" aria-label=\"Direct link to section: 3.5 Explain geometric transformations (reflection, translation, rotation and dilation) which is connected to contextual problems.4.5 Resolve contextual problems related to geometric transformation (reflection, translation, rotation and dilation)\"\u003e\u003c/a\u003e3.5 Explain geometric transformations (reflection, translation, rotation and dilation) which is connected to contextual problems.4.5 Resolve contextual problems related to geometric transformation (reflection, translation, rotation and dilation)\u003c/h2\u003e\u003ch2 id=\"b-learning-objectives\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-learning-objectives\" class=\"header-mark\" aria-label=\"Direct link to section: B. Learning Objectives\"\u003e\u003c/a\u003eB. Learning Objectives\u003c/h2\u003e\u003ch2 id=\"after-studying-the-geometry-transformation-material-you-are-expected-to-be-able-to\" class=\"headerLink\"\u003e\n \u003ca href=\"#after-studying-the-geometry-transformation-material-you-are-expected-to-be-able-to\" class=\"header-mark\" aria-label=\"Direct link to section: After studying the Geometry Transformation material, you are expected to be able to:\"\u003e\u003c/a\u003eAfter studying the Geometry Transformation material, you are expected to be able to:\u003c/h2\u003e\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eExplain geometric transformations (reflection, translation, rotation and dilation) which is connected to contextual problems.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eResolve contextual problems related to geometric transformation (reflection, translation, rotation and dilation)\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-material\" class=\"header-mark\" aria-label=\"Direct link to section: C. Material\"\u003e\u003c/a\u003eC. Material\u003c/h2\u003e\u003ch2 id=\"transformation-is-a-process-that-changes-the-position-or-shape-of-an-object-mathematically-transformation-of-a-flat-object-means-moving-or-mapping-all-points-on-an-object-to-other-places-on-the-same-plane-there-are-transformations-that-change-the-position-of-objectsbuildings-but-their-shape-and-size-are-the-same-as-before-the-transformation-there-are-also-transformations-that-change-the-position-and-size-of-the-objectshape-being-transformed-even-though-the-shape-is-still-the-same-the-type-of-transformation-we-will-study-is-mirroring-reflection-shift-translation-rotation-rotation-and-multiplication-dilatation\" class=\"headerLink\"\u003e\n \u003ca href=\"#transformation-is-a-process-that-changes-the-position-or-shape-of-an-object-mathematically-transformation-of-a-flat-object-means-moving-or-mapping-all-points-on-an-object-to-other-places-on-the-same-plane-there-are-transformations-that-change-the-position-of-objectsbuildings-but-their-shape-and-size-are-the-same-as-before-the-transformation-there-are-also-transformations-that-change-the-position-and-size-of-the-objectshape-being-transformed-even-though-the-shape-is-still-the-same-the-type-of-transformation-we-will-study-is-mirroring-reflection-shift-translation-rotation-rotation-and-multiplication-dilatation\" class=\"header-mark\" aria-label=\"Direct link to section: Transformation is a process that changes the position or shape of an object. Mathematically, transformation of a flat object means moving or mapping all points on an object to other places on the same plane. There are transformations that change the position of objects/buildings, but their shape and size are the same as before the transformation. There are also transformations that change the position and size of the object/shape being transformed even though the shape is still the same. The type of transformation we will study is mirroring (reflection), shift (translation), rotation (rotation), and multiplication (dilatation).\"\u003e\u003c/a\u003eTransformation is a process that changes the position or shape of an object. Mathematically, transformation of a flat object means moving or mapping all points on an object to other places on the same plane. There are transformations that change the position of objects/buildings, but their shape and size are the same as before the transformation. There are also transformations that change the position and size of the object/shape being transformed even though the shape is still the same. The type of transformation we will study is mirroring (reflection), shift (translation), rotation (rotation), and multiplication (dilatation).\u003c/h2\u003e\u003ch3 id=\"reflection\" class=\"headerLink\"\u003e\n \u003ca href=\"#reflection\" class=\"header-mark\" aria-label=\"Direct link to section: Reflection\"\u003e\u003c/a\u003eReflection\u003c/h3\u003e\u003cp\u003eReflection or mirroring is a type of transformation that moves each point on a plane (or geometric shapes) by using the properties of the object and its image in a plane mirror. Look at the image below:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGFpgPFg0FjucAIONouuThhVegWJydf9v6jyOjBZnXE_aJuH7BYc8He4GtmUR4BKdxapzstWLKaas9NBWUaareW1J1OXjAVwVugwgVqiFp28YQfkqdhc0Mp2zyBGFMVhCr06askg2Vy2Q/s191/refleksi2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"166\" data-original-width=\"191\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGFpgPFg0FjucAIONouuThhVegWJydf9v6jyOjBZnXE_aJuH7BYc8He4GtmUR4BKdxapzstWLKaas9NBWUaareW1J1OXjAVwVugwgVqiFp28YQfkqdhc0Mp2zyBGFMVhCr06askg2Vy2Q/s0/refleksi2.png\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eDo you remember looking in the mirror? When you approach the mirror, it appears that your reflection will also approach the mirror. When you move away from the mirror, your image will also move away from the mirror. The characteristics of the image of an object formed by mirroring include the following. - The reflected image of a shape has the same shape and size as the original shape. - The distance of the image to the mirror is the same as the distance of the original object to the mirror. - The image of the shape in the mirror is opposite to the original shape.**\u003c/p\u003e\n\u003ch3 id=\"various-reflections\" class=\"headerLink\"\u003e\n \u003ca href=\"#various-reflections\" class=\"header-mark\" aria-label=\"Direct link to section: Various reflections\"\u003e\u003c/a\u003eVarious reflections\u003c/h3\u003e\u003cp\u003e**\u003c/p\u003e\n\u003ch4 id=\"1-reflection-on-the-x-axis\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-reflection-on-the-x-axis\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Reflection on the x-axis\"\u003e\u003c/a\u003e1. Reflection on the x-axis\u003c/h4\u003e\u003cp\u003eExample: Look for the reflection of point A(-2, 3) to the x-axisSolution:Look at the following picture:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2mlgwr1Xd7U-4EL789FsiPDKxiulcvF5Qzl08VHdLXHFjB4CF_Ya3bdL-gD3_sM6B5_JiFejgDH0-8fU6AnIV_hUnJwwAYwVDVWgaxJdi2AzsqNjctxqrBsbLnexpgN7B8X05et5CJKc/s407/refleksi3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"405\" data-original-width=\"407\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2mlgwr1Xd7U-4EL789FsiPDKxiulcvF5Qzl08VHdLXHFjB4CF_Ya3bdL-gD3_sM6B5_JiFejgDH0-8fU6AnIV_hUnJwwAYwVDVWgaxJdi2AzsqNjctxqrBsbLnexpgN7B8X05et5CJKc/s320/refleksi3.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch4 id=\"2-reflection-on-the-y-axis\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-reflection-on-the-y-axis\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Reflection on the y-axis\"\u003e\u003c/a\u003e2. Reflection on the y-axis\u003c/h4\u003e\u003cp\u003eExample: Look for the reflection of point A(-2, 3) towards the y-axisSolution:Look at the following picture:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi65bxHI5sg9w14LuFKkclIj2OdWljahmHsxyY_-T3H2TbMUAqi-X3cD_lwxz1QYwp3KCLXOkpsr5RUYjfqhsQBKmF9yww-g86SI0uee1pyP8m1Tryg-OSiC3Q2InZChoaL6bkKa-Mo0Bg/s405/refleksi4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"405\" data-original-width=\"303\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi65bxHI5sg9w14LuFKkclIj2OdWljahmHsxyY_-T3H2TbMUAqi-X3cD_lwxz1QYwp3KCLXOkpsr5RUYjfqhsQBKmF9yww-g86SI0uee1pyP8m1Tryg-OSiC3Q2InZChoaL6bkKa-Mo0Bg/s320/refleksi4.png\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch4 id=\"3-reflection-on-the-line-y--x\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-reflection-on-the-line-y--x\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Reflection on the line y = x\"\u003e\u003c/a\u003e3. Reflection on the line y = x\u003c/h4\u003e\u003cp\u003eExample: Look for the reflection of point A(-2, 3) to line y = xSolution:Look at the following picture:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGwVGDBN31L5ip0ZrQQShBE7N8ErW30hjZK4OvdhU4FCMusH1gu5p1WT8fCdtlEk2jtchmcpzs5HGSrttXC83-oFQJXTJrawcJxY0l__RBilIh89_thNO7sPkDTbHjbs7wpXu5cBLTWdc/s426/refleksi5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"411\" data-original-width=\"426\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGwVGDBN31L5ip0ZrQQShBE7N8ErW30hjZK4OvdhU4FCMusH1gu5p1WT8fCdtlEk2jtchmcpzs5HGSrttXC83-oFQJXTJrawcJxY0l__RBilIh89_thNO7sPkDTbHjbs7wpXu5cBLTWdc/s320/refleksi5.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch4 id=\"4-reflection-on-the-line-y---x\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-reflection-on-the-line-y---x\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Reflection on the line y = -x\"\u003e\u003c/a\u003e4. Reflection on the line y = -x\u003c/h4\u003e\u003cp\u003eExample: Look for the reflection of point A(-2, 3) to line y = -xSolution:Look at the following picture:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia20SeZcjWuxVBAMxBrj2qrb094_qF4KdS5FFD0lBNjU72bxRFCP438kfWKHNu2xErW9CIaVwFZ189uNHlsWHKNMS_PU3_Q28s_JI78_p7B0QOxCR6afuymvZeE-Hwuw8CAFlbC0Gqy5o/s411/refleksi6.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"408\" data-original-width=\"411\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia20SeZcjWuxVBAMxBrj2qrb094_qF4KdS5FFD0lBNjU72bxRFCP438kfWKHNu2xErW9CIaVwFZ189uNHlsWHKNMS_PU3_Q28s_JI78_p7B0QOxCR6afuymvZeE-Hwuw8CAFlbC0Gqy5o/s320/refleksi6.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eBelow we provide a simulation of the reflection of triangle ABC on the x-axis, please try it.\u003c/p\u003e\n\u003ciframe height=\"500px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/dmxpvhyn/width/700/height/500/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Refleksi segitiga ABC\" width=\"680px\"\u003e \u003c/iframe\u003e\n\u003cp\u003eBased on the examples above, the reflection results of a point can be concluded as presented in the following table:**\u003c/p\u003e\n\u003cstyle type=\"text/css\"\u003e\n.tg {border-collapse:collapse;border-color:#C44D58;border-spacing:0;}\n.tg td{background-color:#F9CDAD;border-color:#C44D58;border-style:solid;border-width:1px;color:#002b36;\n font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg th{background-color:#FE4365;border-color:#C44D58;border-style:solid;border-width:1px;color:#fdf6e3;\n font-family:Arial, sans-serif;font-size:14px;font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg .tg-baqh{text-align:center;vertical-align:top}\n.tg .tg-bw1t{background-color:#FFA4A0;text-align:left;vertical-align:top}\n.tg .tg-of00{background-color:#FFA4A0;font-style:italic;text-align:left;vertical-align:top}\n.tg .tg-0lax{text-align:left;vertical-align:top}\n.tg .tg-8zwo{font-style:italic;text-align:left;vertical-align:top}\n@media screen and (max-width: 767px) {.tg {width: auto !important;}.tg col {width: auto !important;}.tg-wrap {overflow-x: auto;-webkit-overflow-scrolling: touch;}}\u003c/style\u003e\n\u003cp\u003e**\u003c/p\u003e\n\u003cdiv class=\"tg-wrap\"\u003e\n\u003cstyle type=\"text/css\"\u003e\n.tg {border-collapse:collapse;border-color:#93a1a1;border-spacing:0;}\n.tg td{background-color:#fdf6e3;border-color:#93a1a1;border-style:solid;border-width:1px;color:#002b36;\n font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg th{background-color:#657b83;border-color:#93a1a1;border-style:solid;border-width:1px;color:#fdf6e3;\n font-family:Arial, sans-serif;font-size:14px;font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg .tg-c9xk{background-color:#fe0000;border-color:#ffccc9;color:#3531ff;text-align:center;vertical-align:top}\n.tg .tg-alz1{background-color:#eee8d5;text-align:left;vertical-align:top}\n.tg .tg-m4ky{background-color:#eee8d5;font-style:italic;text-align:left;vertical-align:top}\n.tg .tg-0lax{text-align:left;vertical-align:top}\n.tg .tg-8zwo{font-style:italic;text-align:left;vertical-align:top}\n.tg .tg-9qu3{background-color:#dae8fc;font-style:italic;text-align:left;vertical-align:top}\n@media screen and (max-width: 767px) {.tg {width: auto !important;}.tg col {width: auto !important;}.tg-wrap {overflow-x: auto;-webkit-overflow-scrolling: touch;}}\u003c/style\u003e\n\u003c/div\u003e\n\u003cdiv class=\"tg-wrap\"\u003e\n\u003cstyle type=\"text/css\"\u003e\n.tg {border-collapse:collapse;border-color:#93a1a1;border-spacing:0;}\n.tg td{background-color:#fdf6e3;border-color:#93a1a1;border-style:solid;border-width:1px;color:#002b36;\n font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg th{background-color:#657b83;border-color:#93a1a1;border-style:solid;border-width:1px;color:#fdf6e3;\n font-family:Arial, sans-serif;font-size:14px;font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;}\n.tg .tg-q3w6{background-color:#68cbd0;border-color:#ffccc9;color:#3531ff;text-align:center;vertical-align:top}\n.tg .tg-alz1{background-color:#eee8d5;text-align:left;vertical-align:top}\n.tg .tg-0lax{text-align:left;vertical-align:top}\n.tg .tg-zf58{background-color:#dae8fc;text-align:left;vertical-align:top}\n@media screen and (max-width: 767px) {.tg {width: auto !important;}.tg col {width: auto !important;}.tg-wrap {overflow-x: auto;-webkit-overflow-scrolling: touch;}}\u003c/style\u003e\n\u003c/div\u003e\n\u003cdiv class=\"tg-wrap\"\u003e\n\u003ctable class=\"tg\"\u003e\n\u003cthead\u003e\n \u003ctr\u003e\n\u003cth class=\"tg-q3w6\"\u003eAxis Line\u003c/th\u003e\n\u003cth class=\"tg-q3w6\"\u003eStarting Point\u003c/th\u003e\n\u003cth class=\"tg-q3w6\"\u003eShadow\u003c/th\u003e\n\u003cth class=\"tg-q3w6\"\u003eExample\u003c/th\u003e\n \u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-0lax\"\u003ex-axis\u003c/td\u003e\n\u003ctd class=\"tg-alz1\"\u003eA (x,y)\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003eA' (x, -y)\u003c/td\u003e\n\u003ctd class=\"tg-alz1\"\u003eA (-2, 3) Become A' (-2, -3)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-0lax\"\u003ey-axis\u003c/td\u003e\n\u003ctd class=\"tg-alz1\"\u003eA (x,y)\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003eA' (-x,y)\u003c/td\u003e\n\u003ctd class=\"tg-alz1\"\u003eA (-2, 3) Become A' (2, 3)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-0lax\"\u003eLine y = x\u003c/td\u003e\n\u003ctd class=\"tg-alz1\"\u003eA (x,y)\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003eA' (y, x)\u003c/td\u003e\n\u003ctd class=\"tg-zf58\"\u003eA (-2, 3) Become A' (3, -2)\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n\u003ctd class=\"tg-0lax\"\u003eLine y = -x\u003c/td\u003e\n\u003ctd class=\"tg-alz1\"\u003eA (x,y)\u003c/td\u003e\n\u003ctd class=\"tg-0lax\"\u003eA' (-y, -x)\u003c/td\u003e\n\u003ctd class=\"tg-alz1\"\u003eA (-2, 3) Become A' (-3, 2)\u003c/td\u003e\n \u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003ch2 id=\"d-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: D. Evaluation\"\u003e\u003c/a\u003eD. Evaluation\u003c/h2\u003e\u003cp\u003eAfter studying the geometric transformation material in the reflection sub-material, we now carry out an evaluation to find out our understanding of this reflection material. Please do the following quiz:\u003c/p\u003e\n\u003ciframe frameborder=\"0\" height=\"400\" marginheight=\"0\" marginwidth=\"0\" src=\"https://docs.google.com/forms/d/e/1FAIpQLSfxwg_X0EalTEu62R6pS4cst6jhPsMtMzxpFbcZt0OJSh6BUA/viewform?embedded=true\" width=\"680\"\u003eMemuat…\u003c/iframe\u003e\n\u003cp\u003eThis is the geometric transformation material in the reflection submaterial. I hope this is helpful. 😊🙏\u003c/p\u003e\n", "language": "en" }, { "title" : "FUNCTION part 2", "date_published" : "2020-09-28T15:52:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/fungsi-part-2/", "url" : "https://www.aantriono.com/en/fungsi-part-2/", "summary": " Domain, Codomain, and Range of Functions\nFunction material part 2 is a continuation of function material part 1, in which we will learn about functions in the domain, codomain, range, function notation and determining the value of a function. Let\u0026rsquo;s just look at the material on the broadcast following slides:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"domain-codomain-and-range-of-functions\" class=\"headerLink\"\u003e\n \u003ca href=\"#domain-codomain-and-range-of-functions\" class=\"header-mark\" aria-label=\"Direct link to section: Domain, Codomain, and Range of Functions\"\u003e\u003c/a\u003eDomain, Codomain, and Range of Functions\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAQ99R7RJS3YTNvAW3cX-gJ0WVilIP7mBNGRa54I0yBSjPt0ft7fRhIu3qW6yeMfFH9e5TZwwxioepNpu8BWZgQa34B0COUpHlw5XImtk-FGCH8l1laDbSXfCRt537r-J0r-SZdZW3QsA/s587/fungsi1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"364\" data-original-width=\"587\" height=\"396\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAQ99R7RJS3YTNvAW3cX-gJ0WVilIP7mBNGRa54I0yBSjPt0ft7fRhIu3qW6yeMfFH9e5TZwwxioepNpu8BWZgQa34B0COUpHlw5XImtk-FGCH8l1laDbSXfCRt537r-J0r-SZdZW3QsA/w640-h396/fungsi1.png\" title=\"Gambar materi relasi dan fungsi\" width=\"640\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eFunction material part 2 is a continuation of function material part 1, in which we will learn about functions in the domain, codomain, range, function notation and determining the value of a function. Let\u0026rsquo;s just look at the material on the broadcast following slides:\u003c/p\u003e\n\u003c!--more--\u003e\n\u003ciframe allowfullscreen=\"true\" frameborder=\"0\" height=\"749\" mozallowfullscreen=\"true\" src=\"https://docs.google.com/presentation/d/e/2PACX-1vRDKF9ErmBHsjXZ02Mvy_iFLQ9nrqBXToVE4bOCIGa7AeNXolBELLNMziI4lCigzA/embed?start=true\u0026loop=true\u0026delayms=3000\" webkitallowfullscreen=\"true\" width=\"1280\"\u003e\u003c/iframe\u003e\n\u003cp\u003eTo understand the material in the slide show more clearly, please watch the video explaining the material below:\u003c/p\u003e\n\u003ciframe allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\" frameborder=\"0\" height=\"315\" src=\"https://www.youtube.com/embed/JpIkGZCdXS4\" width=\"650\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"learning-evaluation-with-moodle\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-evaluation-with-moodle\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Evaluation with Moodle\"\u003e\u003c/a\u003eLearning Evaluation with Moodle\u003c/h2\u003e\u003cp\u003eBecause the material about relations and functions has been completed, next we will carry out an evaluation to find out the extent of knowledge about the material about relations and functions. Please open the moodle by:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003eClick the three line icon in the top left corner\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eClick E-Learning\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eClick My MoodleThen an image will appear as below, then enter your respective username and password.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-NNsshehJybnINpQ4cmxOSfpmI7RLOSbaOBBl8Q3F-T2Zc0z2ic0ywq0UAN1wVB1ZDwSRQLbY0QhEC7pw2WYIp5GowL5VJuR6BctPltgURpJmbBv2n_7nVs2qLNsi7m6gsc6Wv1mdrxQ/s623/moodle.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"336\" data-original-width=\"623\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-NNsshehJybnINpQ4cmxOSfpmI7RLOSbaOBBl8Q3F-T2Zc0z2ic0ywq0UAN1wVB1ZDwSRQLbY0QhEC7pw2WYIp5GowL5VJuR6BctPltgURpJmbBv2n_7nVs2qLNsi7m6gsc6Wv1mdrxQ/s320/moodle.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThat\u0026rsquo;s the material for part 2 of this function, I hope it\u0026rsquo;s useful. Good luck for the future. Thank you. 😊🙏\u003c/p\u003e", "language": "en" }, { "title" : "FUNCTION part 1", "date_published" : "2020-09-27T08:02:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/fungsi/", "url" : "https://www.aantriono.com/en/fungsi/", "summary": " Introduction\nDefinition of a FunctionA function from set AAA to set BBB is a special relation that maps every member of set AAA to exactly one member of set BBB.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmBT875wnTtQ1OjeM872tKa1LFlZvm2EZ2OTaBZ45O5xePgx4gLnAGbBdAgbuN0oT_P021_1zwOsFzTj98-T5v3kdmFeAb-vvYK0i1zFBgaoy3dD_ZFcGq1m3U5PZbnmuK4gSlaqmB150/w640-h460/pemetaan1.png' alt=\"Grade 8 function material\" \u003e\u003c/p\u003e\n\u003ch2 id=\"definition-of-a-function\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-of-a-function\" class=\"header-mark\" aria-label=\"Direct link to section: Definition of a Function\"\u003e\u003c/a\u003eDefinition of a Function\u003c/h2\u003e\u003cp\u003eA function from set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a special relation that maps \u003cstrong\u003eevery member of set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/strong\u003e to \u003cstrong\u003eexactly one member of set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eCharacteristics of a function:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eevery member of set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e must have a partner,\u003c/li\u003e\n\u003cli\u003eevery member of set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e has only one partner.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"examples-of-functions-and-non-functions\" class=\"headerLink\"\u003e\n \u003ca href=\"#examples-of-functions-and-non-functions\" class=\"header-mark\" aria-label=\"Direct link to section: Examples of Functions and Non-Functions\"\u003e\u003c/a\u003eExamples of Functions and Non-Functions\u003c/h2\u003e\u003cp\u003eObserve the following figures to distinguish a function from a non-function.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5uQx-X4LojQrZhQ_oaxE9-Zqoap2uKbSP0F_edBu-stE8Ewy-kCKELKnM9l7AmlQ9Y4KDcKT-FjR52ux9WkmA_h-TibGXOqC4Hh4jHNoiyO0gKdUPK34xovYrsLdNi1tOzXXONtezZl8/s772/pemetaan+2.png' alt=\"Function and non-function example 1\" \u003e\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBD1kntXjfBfnZXOjrOLwJrosEau6f21Q3cMRFmx19xuAIOMaql1w5eVy3R9oigl5T4RNbZYPcGmbSw9otGU_g2DWrr8socWgDSHnCvQ-TPqG0UY_uDSGJPhyR0jKHSgDKXgJkEZHTuM4/s624/pemetaan3.png' alt=\"Function and non-function example 2\" \u003e\u003c/p\u003e\n\u003ch2 id=\"determining-the-number-of-possible-functions\" class=\"headerLink\"\u003e\n \u003ca href=\"#determining-the-number-of-possible-functions\" class=\"header-mark\" aria-label=\"Direct link to section: Determining the Number of Possible Functions\"\u003e\u003c/a\u003eDetermining the Number of Possible Functions\u003c/h2\u003e\u003ch3 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h3\u003e\u003cp\u003eDetermine the number of possible mappings from:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA=\\{1,2\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eto:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB=\\{a,b\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eObserve the illustration below.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTt_RQTLIDjt0ePkOEQL7wAgTqwdP1GJEuzMwvg56x18EsrEclDL1yRLkD-nXH31vlMl0sIZvmY59KqQCPcZAlzB07SFDVVWkWzzi-iuyfk7xVoTADdDSDEj9rOQ6SwrAsyt5M4rTmSII/s1110/pemetaan4.png' alt=\"Mapping from A to B\" \u003e\u003c/p\u003e\n\u003cp\u003eSince each member of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e can be paired with two choices in \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, the number of possible functions is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2^2=4\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the number of possible mappings from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{1,2\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{a,b\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cstrong\u003e4\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h3\u003e\u003cp\u003eDetermine the number of possible mappings from:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA=\\{1,2,3\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eto:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB=\\{a,b\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eObserve the following figure.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZk4_V7iW4VsqI-coS0jDmJtaz4ajsfrKJYgvtqJIy6bPqOYru2mN3YGn1cf9t9tYEQ1uHHSXo_gaSwAOSGOUM240rrL3K4spPCuxlhWMb0JC792lrBnX8fU02zw5TLt0QXE25oeU8R64/s624/pemetaan5.png' alt=\"Three elements mapped to two elements\" \u003e\u003c/p\u003e\n\u003cp\u003eThe number of possible functions is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2^3=8\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the number of possible mappings from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA=\\{1,2,3\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB=\\{a,b\\}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cstrong\u003e8\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"conclusion\" class=\"headerLink\"\u003e\n \u003ca href=\"#conclusion\" class=\"header-mark\" aria-label=\"Direct link to section: Conclusion\"\u003e\u003c/a\u003eConclusion\u003c/h2\u003e\u003cp\u003eIf the number of members of set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(A)=a\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and the number of members of set \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(B)=b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ethe number of possible functions from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nb^a\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7144em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003ethe number of possible functions from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^b\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8991em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8991em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eThe number of possible mappings from:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nA=\\{2,3,4\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eto:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e{\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e16\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e}\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nB=\\{12,15,16,20\\}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e{\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e16\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mclose\"\u003e}\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eis \u0026hellip;\u003c/p\u003e\n\u003cp\u003eKnown:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(A)=3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en(B)=4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eAsked: the number of possible mappings from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eSolution:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eA\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e64\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nn(B)^{n(A)}=4^3=64\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.188em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.938em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003cspan class=\"mopen mtight\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003eA\u003c/span\u003e\u003cspan class=\"mclose mtight\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e64\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the number of possible mappings from \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eA\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eA\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eB\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.05017em;\"\u003eB\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is \u003cstrong\u003e64\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"explanation-video\" class=\"headerLink\"\u003e\n \u003ca href=\"#explanation-video\" class=\"header-mark\" aria-label=\"Direct link to section: Explanation Video\"\u003e\u003c/a\u003eExplanation Video\u003c/h2\u003e\u003cp\u003eTo better understand this material, watch the following video.\u003c/p\u003e\n\u003ciframe allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\" frameborder=\"0\" height=\"315\" src=\"https://www.youtube.com/embed/2-GZOZ3yAe8\" width=\"650\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThe next topic covers function notation, evaluating functions, determining a function rule from known values, and one-to-one correspondence. That material is presented in \u003cstrong\u003eFunction part 2\u003c/strong\u003e.\u003c/p\u003e\n", "language": "en" }, { "title" : "Applications of Quadratic Functions", "date_published" : "2020-09-26T07:37:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/aplikasi-fungsi-kuadrat/", "url" : "https://www.aantriono.com/en/aplikasi-fungsi-kuadrat/", "summary": " Introduction\nMany daily-life problems can be solved using quadratic functions by first expressing them as mathematical models. Once the model is built, the problem can be solved systematically.\nExample Problems Problem 1The sum of two integers is 20. Determine the greatest possible product of the two numbers.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4S57XMeO82kPSGaoQRMZk5wPZFDq3R-qYCqaTOB5OrhT94Cmkltehl-EqD300AWcYo8tBUZpbS9lx2EcO9HbUNrLmD0p93Xd2sWVwi0daZczW5ekrOXndVM0BW510WCCDSRVxjYXQkvY/w640-h360/fungsi-kuadrat.jpg' alt=\"Applications of quadratic functions\" \u003e\u003c/p\u003e\n\u003cp\u003eMany daily-life problems can be solved using quadratic functions by first expressing them as mathematical models. Once the model is built, the problem can be solved systematically.\u003c/p\u003e\n\u003ch2 id=\"example-problems\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-problems\" class=\"header-mark\" aria-label=\"Direct link to section: Example Problems\"\u003e\u003c/a\u003eExample Problems\u003c/h2\u003e\u003ch3 id=\"problem-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#problem-1\" class=\"header-mark\" aria-label=\"Direct link to section: Problem 1\"\u003e\u003c/a\u003eProblem 1\u003c/h3\u003e\u003cp\u003eThe sum of two integers is 20. Determine the greatest possible product of the two numbers.\u003c/p\u003e\n\u003ch3 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eLet the two integers be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ey\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eSince their sum is 20:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx+y=20\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny=20-x\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIf their product is written as a function \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ef(x)\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nf(x) \u0026amp;= x \\cdot y \\\\\n \u0026amp;= x(20-x) \\\\\n \u0026amp;= 20x-x^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5241em;vertical-align:-2.0121em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe maximum value of this quadratic function is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmtext\u003emax\u003c/mtext\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e20\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e400\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e100\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nf_{\\text{max}} \u0026amp;= \\frac{-b^2-4ac}{4a} \\\\\n \u0026amp;= \\frac{-20^2-4(-1)(0)}{4(-1)} \\\\\n \u0026amp;= \\frac{-400}{-4} \\\\\n \u0026amp;= 100\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:9.095em;vertical-align:-4.2975em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7975em;\"\u003e\u003cspan style=\"top:-6.7975em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003emax\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3204em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.7629em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1464em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.2975em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7975em;\"\u003e\u003cspan style=\"top:-6.7975em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3204em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.7629em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e400\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1464em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e100\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.2975em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the greatest possible product is \u003cstrong\u003e100\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"problem-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#problem-2\" class=\"header-mark\" aria-label=\"Direct link to section: Problem 2\"\u003e\u003c/a\u003eProblem 2\u003c/h3\u003e\u003cp\u003eA rocket is launched vertically upward. Its height after \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003et\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6151em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e seconds is given by:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nh(t)=60t-5t^2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eWhat is the maximum height reached by the rocket?\u003c/p\u003e\n\u003ch3 id=\"solution-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-1\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eGiven:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003et\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nh(t)=60t-5t^2,\\quad a=-5,\\quad b=60,\\quad c=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0585em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe maximum height occurs when:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003et\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nt \u0026amp;= \\frac{-b}{2a} \\\\\n \u0026amp;= \\frac{-60}{2(-5)} \\\\\n \u0026amp;= \\frac{-60}{-10} \\\\\n \u0026amp;= 6\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:8.8056em;vertical-align:-4.1528em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.6528em;\"\u003e\u003cspan style=\"top:-6.6528em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003et\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3454em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.7879em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1214em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.1528em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.6528em;\"\u003e\u003cspan style=\"top:-6.6528em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3454em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.7879em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1214em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.1528em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe maximum height is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmtext\u003emax\u003c/mtext\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3600\u003c/mn\u003e\u003cmn\u003e20\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e180\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nh_{\\text{max}} \u0026amp;= \\frac{b^2-4ac}{-4a} \\\\\n \u0026amp;= \\frac{60^2-4(-5)(0)}{-4(-5)} \\\\\n \u0026amp;= \\frac{3600}{20} \\\\\n \u0026amp;= 180\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:9.095em;vertical-align:-4.2975em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7975em;\"\u003e\u003cspan style=\"top:-6.7975em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eh\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.1514em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003emax\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.2371em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6796em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1464em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.2975em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.7975em;\"\u003e\u003cspan style=\"top:-6.7975em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.2371em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6796em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e20\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3600\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1464em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e180\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.2975em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo, the rocket reaches a maximum height of \u003cstrong\u003e180 meters\u003c/strong\u003e.\u003c/p\u003e\n\u003ch3 id=\"problem-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#problem-3\" class=\"header-mark\" aria-label=\"Direct link to section: Problem 3\"\u003e\u003c/a\u003eProblem 3\u003c/h3\u003e\u003cp\u003eA rectangle has a perimeter of 120 cm. Determine its dimensions so that its area is maximum.\u003c/p\u003e\n\u003ch3 id=\"solution-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-2\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eLet the length be \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e. Then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmtext\u003elength\u003c/mtext\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmtext\u003ewidth\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003eperimeter\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmtext\u003ewidth\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e120\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmtext\u003ewidth\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003ewidth\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{length} + \\text{width} \u0026amp;= \\frac{1}{2}\\times \\text{perimeter} \\\\\nx + \\text{width} \u0026amp;= \\frac{1}{2}\\times 120 \\\\\nx + \\text{width} \u0026amp;= 60 \\\\\n\\text{width} \u0026amp;= 60-x\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:7.6149em;vertical-align:-3.5574em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0574em;\"\u003e\u003cspan style=\"top:-6.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003elength\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ewidth\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.75em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ewidth\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.924em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ewidth\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.424em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ewidth\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.0574em;\"\u003e\u003cspan style=\"top:-6.0574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eperimeter\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.75em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e120\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.924em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.424em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:3.5574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe area is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmtext\u003elength\u003c/mtext\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmtext\u003ewidth\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL(x) \u0026amp;= \\text{length}\\times \\text{width} \\\\\n \u0026amp;= x(60-x) \\\\\n \u0026amp;= 60x-x^2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5241em;vertical-align:-2.0121em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003elength\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003ewidth\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eFrom this function, we get \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=-1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=60\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eIts axis of symmetry is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmfrac\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx \u0026amp;= -\\frac{b}{2a} \\\\\n \u0026amp;= \\frac{-60}{2(-1)} \\\\\n \u0026amp;= \\frac{-60}{-2} \\\\\n \u0026amp;= 30\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:8.8056em;vertical-align:-4.1528em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.6528em;\"\u003e\u003cspan style=\"top:-6.6528em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3454em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.7879em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1214em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.1528em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.6528em;\"\u003e\u003cspan style=\"top:-6.6528em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3714em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3454em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.7879em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.1214em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3714em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.1528em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe maximum area is:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmi\u003eL\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1800\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e900\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e900\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nL(30) \u0026amp;= 60(30)-(30)^2 \\\\\n \u0026amp;= 1800-900 \\\\\n \u0026amp;= 900\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5241em;vertical-align:-2.0121em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eL\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5121em;\"\u003e\u003cspan style=\"top:-4.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1800\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e900\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e900\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0121em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eLength\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmtext\u003eWidth\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e60\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e30\u003c/mn\u003e\u003cmtext\u003e cm\u003c/mtext\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\text{Length} \u0026amp;= 30 \\text{ cm} \\\\\n\\text{Width} \u0026amp;= 60-30=30 \\text{ cm}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3em;vertical-align:-1.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eLength\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eWidth\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.75em;\"\u003e\u003cspan style=\"top:-3.91em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.41em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e60\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e30\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003e cm\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.25em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThe rectangle with maximum area has dimensions \u003cstrong\u003e30 cm × 30 cm\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"learning-video\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-video\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Video\"\u003e\u003c/a\u003eLearning Video\u003c/h2\u003e\u003cp\u003eTo better understand the worked examples above, study the following video.\u003c/p\u003e\n\u003ciframe allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\" frameborder=\"0\" height=\"350\" src=\"https://www.youtube.com/embed/JzRzjJT6mhs\" width=\"680\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThose are examples of quadratic function applications in daily life.\u003c/p\u003e\n", "language": "en" }, { "title" : "Quadratic Functions", "date_published" : "2020-09-25T03:38:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/fungsi-kuadrat/", "url" : "https://www.aantriono.com/en/fungsi-kuadrat/", "summary": " Introduction\nBasic Competencies3.2 Explain quadratic functions using tables, equations, and graphs.\n3.3 Explain the relationship between the coefficients and discriminant of a quadratic function and its graph.\n4.2 Solve problems related to quadratic equations.\n4.3 Present quadratic functions using tables, equations, and graphs.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6dIQGg5mfOOwqOCMnDBC9ny0Txm9HfdNcgSEiZE_5r-9X6uCBMw1yedFC_HcD3_glQegbp5FULCmgpxQtmR7rdO-w1WVWpS5xIMFH7XSwbSOV3-zWfaiH86icZJySce_idPW401CLtf8/w640-h490/fungsi+l.png' alt=\"Quadratic function\" \u003e\u003c/p\u003e\n\u003ch3 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h3\u003e\u003cp\u003e3.2 Explain quadratic functions using tables, equations, and graphs.\u003c/p\u003e\n\u003cp\u003e3.3 Explain the relationship between the coefficients and discriminant of a quadratic function and its graph.\u003c/p\u003e\n\u003cp\u003e4.2 Solve problems related to quadratic equations.\u003c/p\u003e\n\u003cp\u003e4.3 Present quadratic functions using tables, equations, and graphs.\u003c/p\u003e\n\u003cp\u003e4.4 Present and solve contextual problems using the properties of quadratic functions.\u003c/p\u003e\n\u003ch2 id=\"quadratic-function-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#quadratic-function-material\" class=\"header-mark\" aria-label=\"Direct link to section: Quadratic Function Material\"\u003e\u003c/a\u003eQuadratic Function Material\u003c/h2\u003e\u003cp\u003eA quadratic function has the form:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\ny=ax^2+bx+c\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\neq 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi mathvariant=\"double-struck\"\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ex,y \\in \\mathbb{R}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.03588em;\"\u003ey\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6889em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathbb\"\u003eR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eIt can also be written as:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ef\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nf(x)=ax^2+bx+c\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\" style=\"margin-right:0.10764em;\"\u003ef\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eIn studying quadratic functions, common topics include:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ehow to draw the graph,\u003c/li\u003e\n\u003cli\u003eaxis of symmetry,\u003c/li\u003e\n\u003cli\u003ezeros of the function,\u003c/li\u003e\n\u003cli\u003eintercepts,\u003c/li\u003e\n\u003cli\u003evertex,\u003c/li\u003e\n\u003cli\u003ethe effect of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e on the graph.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eFor a more complete explanation, study the slide deck below.\u003c/p\u003e\n\u003ch2 id=\"slide-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#slide-material\" class=\"header-mark\" aria-label=\"Direct link to section: Slide Material\"\u003e\u003c/a\u003eSlide Material\u003c/h2\u003e\u003ciframe allowfullscreen=\"true\" frameborder=\"0\" height=\"500\" mozallowfullscreen=\"true\" src=\"https://docs.google.com/presentation/d/e/2PACX-1vTxHdW3hjU8ey78G7dRozn7bMCL6S76yBTFR1MTLlORmqrIUI7PihJOhq13UmiN8Q/embed?start=true\u0026loop=true\u0026delayms=3000\" webkitallowfullscreen=\"true\" width=\"650\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"video-lesson\" class=\"headerLink\"\u003e\n \u003ca href=\"#video-lesson\" class=\"header-mark\" aria-label=\"Direct link to section: Video Lesson\"\u003e\u003c/a\u003eVideo Lesson\u003c/h2\u003e\u003cp\u003eTo make the material easier to understand, watch the following explanation video.\u003c/p\u003e\n\u003ciframe allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\" frameborder=\"0\" height=\"315\" src=\"https://www.youtube.com/embed/u-4Dc4TxHPA\" width=\"650\"\u003e\u003c/iframe\u003e\n\u003ch2 id=\"graph-simulation\" class=\"headerLink\"\u003e\n \u003ca href=\"#graph-simulation\" class=\"header-mark\" aria-label=\"Direct link to section: Graph Simulation\"\u003e\u003c/a\u003eGraph Simulation\u003c/h2\u003e\u003cp\u003eTo understand the effect of \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e on the graph of a quadratic function, move the values in the simulation below and observe the changes.\u003c/p\u003e\n\u003ciframe height=\"400px\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/zmvkrhjx/width/700/height/450/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false\" style=\"border: 0px;\" title=\"Quadratic function graph simulation\" width=\"700px\"\u003e\u003c/iframe\u003e\n\u003cp\u003eUse a computer or laptop for a better simulation experience.\u003c/p\u003e\n\u003ch2 id=\"download-material\" class=\"headerLink\"\u003e\n \u003ca href=\"#download-material\" class=\"header-mark\" aria-label=\"Direct link to section: Download Material\"\u003e\u003c/a\u003eDownload Material\u003c/h2\u003e\u003cp\u003eReaders who want to download the quadratic function material in PPT format can use the following link:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003ca href=\"https://drive.google.com/file/d/1GMUqflswqjEokbuJoY1a1ucwVLaOU9-w/view?usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eDownload quadratic function material (PPT)\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThat concludes this quadratic function material.\u003c/p\u003e\n", "language": "en" }, { "title" : "Death as a Transition", "date_published" : "2020-09-22T14:53:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kematian-sebagai-suatu-peralihan/", "url" : "https://www.aantriono.com/en/kematian-sebagai-suatu-peralihan/", "summary": " Reflection on the Reality of Death\nIn navigating life in this world, humans certainly experience periods of transition that come and go one after another: there is morning and evening, day and night, health and sickness, life and death. All these things will happen to all of us.\nThere is something that sometimes we forget to think about, even though one day we will face it. That matter is none other than death. Anyone will experience death. Death is an inevitability experienced by every human being, even though the causes are different.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"reflection-on-the-reality-of-death\" class=\"headerLink\"\u003e\n \u003ca href=\"#reflection-on-the-reality-of-death\" class=\"header-mark\" aria-label=\"Direct link to section: Reflection on the Reality of Death\"\u003e\u003c/a\u003eReflection on the Reality of Death\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIL75eEUIEsmuY8G-KldI1BTjgOJp978w4SXLtd9NWRhVKI4UeR9xZMlIN41PFWNyo68LRZu4qSK1cXQaT-35V7vSO7FF0jRvGa8PNaTt6VxGGUHp2s146NLAyOJWkZlVx3hOQFT93W34/w400-h225/WhatsApp+Image+2020-09-22+at+14.36.32.jpeg' alt=\"illustration of death\" \u003e\u003c/p\u003e\n\u003cp\u003eIn navigating life in this world, humans certainly experience periods of transition that come and go one after another: there is morning and evening, day and night, health and sickness, life and death. All these things will happen to all of us.\u003c/p\u003e\n\u003cp\u003eThere is something that sometimes we forget to think about, even though one day we will face it. That matter is none other than death. Anyone will experience death. Death is an inevitability experienced by every human being, even though the causes are different.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003cp\u003eAllah says in the Qur\u0026rsquo;an:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eقُلْ اِنَّ الْمَوْتَ الَّذِى تَفِرُّوْنَ مِنْهُ فَإِنَّهُ مُلَقِيكُمْ ثُمَّ تُرَدُّوْنَ اِلَى عَلِمِ الْغَيْبِ وَالشَّهَدَةِ فَيُنَبِّئُكُمْ بِمَا كُنْتُمْ تَعْمَلُونَ\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;Say: \u0026lsquo;Indeed, the death from which you flee — indeed, it will meet you. Then you will be returned to the Knower of the unseen and the witnessed, and He will inform you about what you used to do.\u0026rsquo;\u0026rdquo; (QS Al-Jumu\u0026rsquo;ah: 8)\u003c/p\u003e\n\u003ch2 id=\"the-reality-of-worldly-life-and-the-afterlife\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-reality-of-worldly-life-and-the-afterlife\" class=\"header-mark\" aria-label=\"Direct link to section: The Reality of Worldly Life and the Afterlife\"\u003e\u003c/a\u003eThe Reality of Worldly Life and the Afterlife\u003c/h2\u003e\u003cp\u003eThen, what is the meaning of our life in the world? The world is a place where we prepare ourselves for the true life, namely life in the afterlife. As a place of preparation, we must have sufficient preparation because of the long journey that lies ahead. Despite our love for the world and its contents, one day this world will leave us, or we will be the ones left behind. The life of this world is only temporary and ephemeral. True life is the life in which there is no death — the afterlife.\u003c/p\u003e\n\u003cp\u003eIf we may think of it this way: this world is just a stopover, not an eternity. A terminal, a temporary stopping place. We transit in it for a moment, until the specified time. After that we leave and continue our journey again.\u003c/p\u003e\n\u003cp\u003eIf that is the nature of the world, why do we devote so much of our lives to worldly needs? Whether we admit it or not, of the 24 hours we spend in a day, we can say that only a few percent are used to prepare for the afterlife. The rest is drained by activities circulating around the world. Yet we really need to balance the two.\u003c/p\u003e\n\u003ch2 id=\"self-examination-and-good-deeds\" class=\"headerLink\"\u003e\n \u003ca href=\"#self-examination-and-good-deeds\" class=\"header-mark\" aria-label=\"Direct link to section: Self-Examination and Good Deeds\"\u003e\u003c/a\u003eSelf-Examination and Good Deeds\u003c/h2\u003e\u003cp\u003eFrom our current age, have we ever examined ourselves? How much virtue have we accumulated? How many good deeds have we gathered? Or how much disobedience have we committed, how many sins have we engaged in? Of course, only we ourselves know.\u003c/p\u003e\n\u003cp\u003eNow, let us remember Allah\u0026rsquo;s infinite blessings. Every moment blood flows in our bodies. But why do we ignore it all? The heartbeat never stops. We blink countless times a day. We breathe every moment, always enjoying it without ever paying for it, without ever thinking about what would happen if Allah withheld that blessing for just a moment. But because of our ignorant and unjust nature, we — intentionally or not — always take it for granted and forget about it.\u003c/p\u003e\n\u003cp\u003eWe often easily thank someone who has helped us, while we often forget to remember Allah, who always showers us with His blessings. As a result, we will definitely forget the afterlife. From here, the world will always consume our time.\u003c/p\u003e\n\u003ch2 id=\"the-importance-of-remembering-death\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-importance-of-remembering-death\" class=\"header-mark\" aria-label=\"Direct link to section: The Importance of Remembering Death\"\u003e\u003c/a\u003eThe Importance of Remembering Death\u003c/h2\u003e\u003cp\u003eMeanwhile, remembering death will encourage a person to prepare provisions for the journey ahead, to avoid committing acts that lead to disobedience, and to foster piety. In connection with remembering death, the Prophet said:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eاَكْثِرُوا مِنْ ذِكْرِ الْمَوْتِ فَإِنَّهُ يُمَحِّصُ الذُّنُوْبَ وَيُزْهِدُ الدُّنْيَا\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;Multiply the remembrance of death, for it will wipe out sins and cause asceticism in the world.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eIn the Islamic perspective, people who remember death often are considered intelligent. Rasulullah SAW said:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eإِكْيَسُ النَّاسِ اَكْثَرُهُمْ ذِكْرًا لِلْمَوْتِ وَاَشَدُّهُمْ اِسْتِعْدَادًا لَهُ اُوْلَئِكَ هُمُ اْلاَكيَاسُ ذَهَبُوْا بِشَرَفِ الدُّنْيَا وَكَرَامَةِ اْلاَخِرَاةِ\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;The most intelligent people are those who remember death the most, and who prepare the most to face it. They are the truly intelligent ones. And they will go to the afterlife bringing with them the honor of this world and the honor of the afterlife.\u0026rdquo; (HR Ibn Majah)\u003c/p\u003e\n\u003cp\u003eLet us always engage in self-reflection, remembering our own shortcomings, reflecting on the mistakes and sins we have committed, always hoping for forgiveness from Him and repenting to Him — before death comes and takes us without us ever realizing it, without ever wanting it, and without ever expecting it. Because death is a transition from worldly life to the afterlife, and surely every soul will experience it.\u003c/p\u003e\n\u003cp\u003eMay we leave this world in a state of faith and husnul khotimah. Aamiin.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eThis article was written to remember and pray for our teacher, our leader, our motivator, the late Mr. Suharno, S.Pd, Principal of SMPN Purworejo, Padang Ratu District, Central Lampung Province, Lampung. 😭\u003c/em\u003e\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eاَللهُمَّ اغْفِرْلَهُ وَارْحَمْهُ وَعَافِهِ وَاعْفُ عَنْهُ وَاَكْرِمْ نُزُلَهُ وَوَسِّعْ مَدْخَلَهُ وَاغْسِلْهُ بِالْمَاءِ وَالثَّلْجِ وَالْبَرْدِ وَنَقِّهِ مِنَ الْخَطَايَا كَمَا يُنَقَّى الثَّوْبُ اْلاَبْيَضُ مِنَ الدَّنَسِ وَاَبْدِلْهُ دَارًاخَيْرًا مِنْ دَارِهِ وَاَهْلاً خَيْرًا مِنْ اَهْلِهِ وَزَوْجًا خَيْرًا مِنْ زَوْجِهِ وَاَدْخِلْهُ الْجَنَّةَ وَاَعِذْهُ مِنْ عَذَابِ الْقَبْرِ وَفِتْنَتِهِ وَمِنْ عَذَابِ النَّارِ. اَللهُمَّ اغْفِرْ لِحَيِّنَا وَمَيِّتِنَا وَشَاهِدِنَا وَغَائِبِنَا وَصَغِيْرَنَا وَكَبِيْرَنَا وَذَكَرِنَا وَاُنْثَانَا. اَللهُمَّ مَنْ اَحْيَيْتَهُ مِنَّا فَاَحْيِهِ عَلَى اْلاِسْلاَمِ وَمَنْ تَوَفَّيْتَهُ مِنَّا فَتَوَفَّهُ عَلَى اْلاِيْمَانِ. اَللهُمَّ لاَتَحْرِمْنَا اَجْرَهُ وَلاَتُضِلَّنَا بَعْدَهُ بِرَحْمَتِكَ يَآاَرْحَمَ الرَّاحِمِيْنَ. وَالْحَمْدُ ِللهِ رَبِّ الْعَالَمِيْنَ\u003c/span\u003e.\u003c/p\u003e\n\u003c/blockquote\u003e", "language": "en" }, { "title" : "How to Upload an ISpring Quiz into Moodle", "date_published" : "2020-09-21T14:51:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-upload-kuis-ispring-ke-dalam-moodle/", "url" : "https://www.aantriono.com/en/cara-upload-kuis-ispring-ke-dalam-moodle/", "summary": " Integrating iSpring Quizzes with Moodle LMS\nThe iSpring Suite application is a versatile e-Learning authoring tool that allows you to create various types of e-Learning content. With iSpring Suite 9 applications, you can insert quizzes, surveys, interactions, dialogue simulations, or screen recordings directly into PowerPoint presentations. In the Quick Start window, we can select the type of material we want to create, such as a new course, quiz or survey, interaction, dialogue simulation, or screencast, or open material that we have previously created. On this occasion we will provide a tutorial on how to upload quizzes created in iSpring into Moodle.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"integrating-ispring-quizzes-with-moodle-lms\" class=\"headerLink\"\u003e\n \u003ca href=\"#integrating-ispring-quizzes-with-moodle-lms\" class=\"header-mark\" aria-label=\"Direct link to section: Integrating iSpring Quizzes with Moodle LMS\"\u003e\u003c/a\u003eIntegrating iSpring Quizzes with Moodle LMS\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDHeBZmCUmmhs54BSTf-xZuhGwcKF4Jm4HJ7_q2FYInolkgGVKEDO-hc3AXrZlHJEt-IgyuByNT4MwISwHn4GJyc79ESmepoVnm0sMMOeb4IpqxahwMZpqmyowaOtO_MdI7jiObSygM6Q/s662/ispring1.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"526\" data-original-width=\"662\" height=\"254\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDHeBZmCUmmhs54BSTf-xZuhGwcKF4Jm4HJ7_q2FYInolkgGVKEDO-hc3AXrZlHJEt-IgyuByNT4MwISwHn4GJyc79ESmepoVnm0sMMOeb4IpqxahwMZpqmyowaOtO_MdI7jiObSygM6Q/w320-h254/ispring1.png\" title=\"Gambar aplikasi iSpring\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThe iSpring Suite application is a versatile e-Learning authoring tool that allows you to create various types of e-Learning content. With iSpring Suite 9 applications, you can insert quizzes, surveys, interactions, dialogue simulations, or screen recordings directly into PowerPoint presentations. In the Quick Start window, we can select the type of material we want to create, such as a new course, quiz or survey, interaction, dialogue simulation, or screencast, or open material that we have previously created. On this occasion we will provide a tutorial on how to upload quizzes created in iSpring into Moodle.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003ch2 id=\"step-by-step-upload-guide\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-upload-guide\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step Upload Guide\"\u003e\u003c/a\u003eStep-by-Step Upload Guide\u003c/h2\u003e\u003ch2 id=\"step-by-step-upload-guide-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#step-by-step-upload-guide-1\" class=\"header-mark\" aria-label=\"Direct link to section: Step-by-Step Upload Guide\"\u003e\u003c/a\u003eStep-by-Step Upload Guide\u003c/h2\u003e\u003cp\u003eThe steps are as follows:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eAfter you have finished creating the quiz in iSpring, click the publish menu as shown in the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXX4F15O8TxkBJ48ScA6xEKUIDepmdSlkatWDoJ7wR_zLbsybiL54ISO-X3vkfL4ZJvxsqPp2b_jnrVG6NNtpGfQIE1kEt57IP5FAgXnIOOI43ulM2SYGH9UobPiaAwwsQ2HT13a6Zpe0/s800/ispring1.1.2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"682\" data-original-width=\"800\" height=\"273\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXX4F15O8TxkBJ48ScA6xEKUIDepmdSlkatWDoJ7wR_zLbsybiL54ISO-X3vkfL4ZJvxsqPp2b_jnrVG6NNtpGfQIE1kEt57IP5FAgXnIOOI43ulM2SYGH9UobPiaAwwsQ2HT13a6Zpe0/w320-h273/ispring1.1.2.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThen select and click LMS, Write the name of the quiz, Select a storage location, Check use iSpring if you want to open it on your cellphone/tablet using iSpring Play, Select Scorm 1.2 and finally click Publish.\u003c/p\u003e\n\u003col start=\"2\"\u003e\n\u003cli\u003eNext, open the Moodle LMS\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTgR8T1YV2zC-yNIMRC8WG9E6n2RKvwgRijD3RGnFVhOJNWdvbPg2gxg99xOIkg16iH2L_YNGjE18T90XCtmbzRDeRXbDBJ4QooY10ejrBi9vSsdZ31CR9-E9cgNLHIjpN3xdqXD28AGY/s891/ispring2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"621\" data-original-width=\"891\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTgR8T1YV2zC-yNIMRC8WG9E6n2RKvwgRijD3RGnFVhOJNWdvbPg2gxg99xOIkg16iH2L_YNGjE18T90XCtmbzRDeRXbDBJ4QooY10ejrBi9vSsdZ31CR9-E9cgNLHIjpN3xdqXD28AGY/s320/ispring2.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"3\"\u003e\n\u003cli\u003e\n\u003cp\u003eSelect the course you want to add the iSpring quiz to, you can click on the list on the left or the course image as shown below:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz6y6v7OsoaewJ4ZLven-ZFSxhpBMuHJz9jREWaqDwTQVKrCibREBXUhFcU_j8xLy38go4mLGpsLFluMZKbzLGQxE4oZn7stOQVNhWvaQuBxQkUVELUJq4XUZUZqLXr0diqE8Dp0bxAEU/s880/ispring3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"486\" data-original-width=\"880\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz6y6v7OsoaewJ4ZLven-ZFSxhpBMuHJz9jREWaqDwTQVKrCibREBXUhFcU_j8xLy38go4mLGpsLFluMZKbzLGQxE4oZn7stOQVNhWvaQuBxQkUVELUJq4XUZUZqLXr0diqE8Dp0bxAEU/s320/ispring3.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eHere I will add a quiz for class IX\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh32B5Fhqf3xn-AC3dfWrImr8Kbub74T0HD0Jx5xFIkqyhjlZG5wzH3s75VNyAWMXBIqltc5WHJqXtYqGqg2WWJqD6-BqO1F7GfaOFu0UWk49K1qkeojMvo3PiTkVEHIBaU2YjI0bJJLGY/s864/ispring4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"220\" data-original-width=\"864\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh32B5Fhqf3xn-AC3dfWrImr8Kbub74T0HD0Jx5xFIkqyhjlZG5wzH3s75VNyAWMXBIqltc5WHJqXtYqGqg2WWJqD6-BqO1F7GfaOFu0UWk49K1qkeojMvo3PiTkVEHIBaU2YjI0bJJLGY/s320/ispring4.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eNext, click the gear icon and click Turn Editing On.\u003c/p\u003e\n\u003col start=\"5\"\u003e\n\u003cli\u003eSelect the material that will be given the daily quiz/assessment then click the + Add an activity or resource icon at the bottom right.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrOkU1uERoK_NhpP8q4WCYe-q5boFsu4QCP8b1Q2vf5KTozdQg75zZOZmqeKtAqePG9EV0JFeID7bWJW0ZmGBguoYT7VhOrcpOQulQPNMq278svkYMXlIPeiWC6vIMZSGI4Osrju4gOxY/s837/ispring5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"436\" data-original-width=\"837\" height=\"167\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrOkU1uERoK_NhpP8q4WCYe-q5boFsu4QCP8b1Q2vf5KTozdQg75zZOZmqeKtAqePG9EV0JFeID7bWJW0ZmGBguoYT7VhOrcpOQulQPNMq278svkYMXlIPeiWC6vIMZSGI4Osrju4gOxY/w320-h167/ispring5.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"6\"\u003e\n\u003cli\u003eA page appears as shown below:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgg2-R-FpKv7Ia2yKIMBJencoVeepLJOJ00K8OhBgd5HaC0AWvN82GnyAJ5BVULnXGKB9nfnJ3b3JoEi4XUqSvrXgQdqKXtwVELHlNCL6Sm1GtCPIVGixfOtlW28PekusxPaVx07HdCkPE/s614/ispring61.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"614\" data-original-width=\"486\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgg2-R-FpKv7Ia2yKIMBJencoVeepLJOJ00K8OhBgd5HaC0AWvN82GnyAJ5BVULnXGKB9nfnJ3b3JoEi4XUqSvrXgQdqKXtwVELHlNCL6Sm1GtCPIVGixfOtlW28PekusxPaVx07HdCkPE/s320/ispring61.png\" /\u003e\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eClick the Scorm Package button and click add.\u003c/p\u003e\n\u003col start=\"7\"\u003e\n\u003cli\u003eAfter that, the following page will appear\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCOl08ukMEEcFwIP6W7ktjHZD96e0pR01HPI9hUdtFK_BkR4EQIqYimHfyRhqVfTxWgWtRlCzgLSnXH6PZOHXMyLZOiRxeNETbks3cNnOqY133Yqb70eUZAhZhrcSsA7watucNmLgBHno/s917/ispring7.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"451\" data-original-width=\"917\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCOl08ukMEEcFwIP6W7ktjHZD96e0pR01HPI9hUdtFK_BkR4EQIqYimHfyRhqVfTxWgWtRlCzgLSnXH6PZOHXMyLZOiRxeNETbks3cNnOqY133Yqb70eUZAhZhrcSsA7watucNmLgBHno/s320/ispring7.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThe Name box must be filled in, adjust it according to your needs. The description is optional, it may or may not be filled in\u003c/p\u003e\n\u003col start=\"8\"\u003e\n\u003cli\u003eAfter that, scroll down, to the section package click the file icon. Documents that can be uploaded have xml and zip extensions.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn-GXN32bcAJSkEzZ6Og0OEb4jlOYYa67UZNl193Ix39EOGPsnpQ8y1BwbzN7gl9JVoJFfHZeTgPnY3asX9sUyKT0IFPp1diPf0dgc1fxckeAu96Fs_FzhnopMdfEjmG21Ajt38m8JovE/s882/ispring8.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"421\" data-original-width=\"882\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn-GXN32bcAJSkEzZ6Og0OEb4jlOYYa67UZNl193Ix39EOGPsnpQ8y1BwbzN7gl9JVoJFfHZeTgPnY3asX9sUyKT0IFPp1diPf0dgc1fxckeAu96Fs_FzhnopMdfEjmG21Ajt38m8JovE/s320/ispring8.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003col start=\"9\"\u003e\n\u003cli\u003e\n\u003cp\u003eAfter that the following page appears, click upload a file and click choose file as shown in the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja8jB-LIgmG0y5jeyC6KD_7BeI0jQZP4sGg4fMzL_i0aO5iCyy6fJ0aB8s9qjEefYhDn-4xvY8jSQ3aznPg9ysED_go387el0uCzZTaYJhwNgvVYoAZyn4vPGaaSLOgrxAIr5CiMKGAJM/s786/ispring9.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"505\" data-original-width=\"786\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja8jB-LIgmG0y5jeyC6KD_7BeI0jQZP4sGg4fMzL_i0aO5iCyy6fJ0aB8s9qjEefYhDn-4xvY8jSQ3aznPg9ysED_go387el0uCzZTaYJhwNgvVYoAZyn4vPGaaSLOgrxAIr5CiMKGAJM/s320/ispring9.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eFind where the ispring quiz file that was created is stored\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL4GCiQDTzyCDV5HZ5iH4SLXHFQkWksJjvAn4yvNlqyNEYszya5-6dYRkwsu_gOJTp_zchrMIeACQz_BG8m5IIlprFtJCO7um5uf1AW6oxuum0iIKf_CgYYTehPLobonDBm4dzdouh1Os/s648/ispring10.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"426\" data-original-width=\"648\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL4GCiQDTzyCDV5HZ5iH4SLXHFQkWksJjvAn4yvNlqyNEYszya5-6dYRkwsu_gOJTp_zchrMIeACQz_BG8m5IIlprFtJCO7um5uf1AW6oxuum0iIKf_CgYYTehPLobonDBm4dzdouh1Os/s320/ispring10.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eOnce found, click the file and click open\u003c/p\u003e\n\u003col start=\"11\"\u003e\n\u003cli\u003e\n\u003cp\u003eAfter that the file name appears and click upload this file as shown in the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDUwyl4dy6u6RjjfpF9LlbOdHSVEP589lwq4v_clvfNTw8rNnZ82Kxu71mi1HcElW2pU7bBNR4PlQHf9hUuwekuZlN3lRcd5C-LrIuSYcy04C3srNnHQwom9-t7FR4ZvWCjTMakuM-Fcw/s791/ispring11.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"535\" data-original-width=\"791\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDUwyl4dy6u6RjjfpF9LlbOdHSVEP589lwq4v_clvfNTw8rNnZ82Kxu71mi1HcElW2pU7bBNR4PlQHf9hUuwekuZlN3lRcd5C-LrIuSYcy04C3srNnHQwom9-t7FR4ZvWCjTMakuM-Fcw/s320/ispring11.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eAfter the file has been successfully uploaded, the file will appear in the package box as shown in the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKoEghd0QU_klxF26qLJifZKIc1sfI3sy12mYiny_AHDwHbn5OC4cN5gph1spdck_Vl1Vh3uvj_2yaWjkmlhvtCAL4yfxnxJWrmRkdpoar4waiTS1R0hjf-omehQ8rbVGtionKR-ICBuE/s875/ispring12.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"424\" data-original-width=\"875\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKoEghd0QU_klxF26qLJifZKIc1sfI3sy12mYiny_AHDwHbn5OC4cN5gph1spdck_Vl1Vh3uvj_2yaWjkmlhvtCAL4yfxnxJWrmRkdpoar4waiTS1R0hjf-omehQ8rbVGtionKR-ICBuE/s320/ispring12.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eAfter that, scroll down, you will see a display like the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAtc5LvyEA2d3GLdc5lFoq2UFTsll5-rXcrh1K4D9Sv0EBjh6X4S3QSCLkL-2ogPTUjNtwKgCGF_BgT4JtJE3vQkV-aBJwOZc5GK_wrJU5MEf7n363D8hM15s0DPIpOsgt3b8Ija-XnYc/s871/ispring13.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"429\" data-original-width=\"871\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAtc5LvyEA2d3GLdc5lFoq2UFTsll5-rXcrh1K4D9Sv0EBjh6X4S3QSCLkL-2ogPTUjNtwKgCGF_BgT4JtJE3vQkV-aBJwOZc5GK_wrJU5MEf7n363D8hM15s0DPIpOsgt3b8Ija-XnYc/s320/ispring13.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003ein the appearance, availability, grade, attempts management sections are regulated according to needs\u003c/p\u003e\n\u003col start=\"14\"\u003e\n\u003cli\u003e\n\u003cp\u003eScroll down, once all the daily quiz/assessment settings are sufficient, please click save and return to course or save and display as shown in the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipSDOh2HdDnNxHB3xyd37a2E-IGGAqdLdYAtiOU9xThv-l2rx_fKBLPWa8cWzEiFdK3iFgJ-WPYy8ExMcxiGLYOEpf0Y7NZI3MoHdW5-k33VIeDKA8kcBYFXYURogKNpGrSbmRbNxVldY/s851/ispring14.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"414\" data-original-width=\"851\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipSDOh2HdDnNxHB3xyd37a2E-IGGAqdLdYAtiOU9xThv-l2rx_fKBLPWa8cWzEiFdK3iFgJ-WPYy8ExMcxiGLYOEpf0Y7NZI3MoHdW5-k33VIeDKA8kcBYFXYURogKNpGrSbmRbNxVldY/s320/ispring14.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eIf in step 14 you click save and return to course, then the material being given a quiz/assessment will display as shown in the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijQeW79Z77YlZ8BEo0VCxJ2C8uPlZgIwXRr7PEQjb-BQBxKsD88XZYgg-ELaV4DNwAGZnTqxSyH_r6GMdq0AsbQdayh3m67qQyZeYehucf5vGc47wN3AS1QNctHKSGy-1mR5Xmti7RPhI/s901/ispring15.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"435\" data-original-width=\"901\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijQeW79Z77YlZ8BEo0VCxJ2C8uPlZgIwXRr7PEQjb-BQBxKsD88XZYgg-ELaV4DNwAGZnTqxSyH_r6GMdq0AsbQdayh3m67qQyZeYehucf5vGc47wN3AS1QNctHKSGy-1mR5Xmti7RPhI/s320/ispring15.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eEdit settings is used to edit quiz settings in Moodle, hide to display the quiz to students. That\u0026rsquo;s how to upload an iSpring quiz into Moodle. Good luck and good luck. Hopefully this tutorial is useful. Thank you. 🙏.\u003c/p\u003e", "language": "en" }, { "title" : "KSN Questions and Discussion in Mathematics", "date_published" : "2020-09-18T13:56:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/soal-dan-pembahasan-ksn-bidang/", "url" : "https://www.aantriono.com/en/soal-dan-pembahasan-ksn-bidang/", "summary": " About the National Science Competition (KSN)\nThe National Science Competition or commonly abbreviated as KSN is a competition event in the field of science for elementary, middle and high school students throughout Indonesia. Previously, KSN was known as the National Science Olympiad (OSN). This name change occurred in 2020 after the Indonesian Minister of Education and Culture, Nadiem Makarim, established the National Achievement Center which oversees all competitions, competitions and festivals in Indonesia. National Science Competition (KSN) It is hoped that this competition will lead students to master science and technology. This competition is also an important part of equalizing achievement and maximizing the potential of talented and characterful students from all corners of the Unitary State of the Republic of Indonesia. Apart from being a strategy to improve the quality of education, science competitions have a special position at various prestigious international events in the mastery of science by students. So, through this systematic and tiered competition system, it is hoped that the widest possible space will be created for students to explore their abilities in the field of science and reach the peak of their best potential. Maximum achievement will be demonstrated by the birth of qualified and highly competitive science competition champions who are ready to compete at the international level.KSN results are also taken into consideration in the selection process of students who will be prepared to represent Indonesia in international science competitions. Science fields in the competition:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"about-the-national-science-competition-ksn\" class=\"headerLink\"\u003e\n \u003ca href=\"#about-the-national-science-competition-ksn\" class=\"header-mark\" aria-label=\"Direct link to section: About the National Science Competition (KSN)\"\u003e\u003c/a\u003eAbout the National Science Competition (KSN)\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip23cd4xHLo6ObFRQsO5Yr_7b2mm5zPzAa-75CeFglZkLfEJ4ySek_dpVRyPOYvzpO_KjeNmyvPSyiyaZI2AEdnhWiTW1VVndp6QD-7IVvou5chM1kJbneV0bGwNhOo-WdUt8Kx24R49g/s1280/ksn.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"720\" data-original-width=\"1280\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip23cd4xHLo6ObFRQsO5Yr_7b2mm5zPzAa-75CeFglZkLfEJ4ySek_dpVRyPOYvzpO_KjeNmyvPSyiyaZI2AEdnhWiTW1VVndp6QD-7IVvou5chM1kJbneV0bGwNhOo-WdUt8Kx24R49g/s16000/ksn.jpg\" title=\"Gambar Kompetisi Sains Nasional\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eThe National Science Competition or commonly abbreviated as KSN is a competition event in the field of science for elementary, middle and high school students throughout Indonesia. Previously, KSN was known as the National Science Olympiad (OSN). This name change occurred in 2020 after the Indonesian Minister of Education and Culture, Nadiem Makarim, established the National Achievement Center which oversees all competitions, competitions and festivals in Indonesia. National Science Competition (KSN) It is hoped that this competition will lead students to master science and technology. This competition is also an important part of equalizing achievement and maximizing the potential of talented and characterful students from all corners of the Unitary State of the Republic of Indonesia. Apart from being a strategy to improve the quality of education, science competitions have a special position at various prestigious international events in the mastery of science by students. So, through this systematic and tiered competition system, it is hoped that the widest possible space will be created for students to explore their abilities in the field of science and reach the peak of their best potential. Maximum achievement will be demonstrated by the birth of qualified and highly competitive science competition champions who are ready to compete at the international level.KSN results are also taken into consideration in the selection process of students who will be prepared to represent Indonesia in international science competitions. Science fields in the competition:\u003c/p\u003e\n\u003c!--more--\u003e\n\u003col\u003e\u003cli\u003eElementary School Level: Mathematics and Natural Sciences.\u003c/li\u003e\n\u003cli\u003eMiddle school level: Mathematics, science and social studies.\u003c/li\u003e\n\u003cli\u003eHigh school level: Mathematics, Physics, Biology, Chemistry, Astronomy, Computers, Economics, Earth and Geography.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"implementation-stages-of-ksn\" class=\"headerLink\"\u003e\n \u003ca href=\"#implementation-stages-of-ksn\" class=\"header-mark\" aria-label=\"Direct link to section: Implementation Stages of KSN\"\u003e\u003c/a\u003eImplementation Stages of KSN\u003c/h2\u003e\u003cp\u003eImplementation Stages The stages of implementing KSN consist of:\u003c/p\u003e\n\u003col\u003e\u003cli\u003eThe school level selection stage is referred to as the School Level National Science Competition (KSN-S)\u003c/li\u003e\n\u003cli\u003eThe selection stages at the district/city level are referred to as the National Science Competition at the District/City level (KSN-K)\u003c/li\u003e\n\u003cli\u003eThe provincial level selection stage is referred to as the National Science Competition at the Provincial level (KSN-P)\u003c/li\u003e\n\u003cli\u003eThe national level selection stage is referred to as the National Science Competition (KSN)\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eSource : \u003ca href=\"https://id.wikipedia.org/wiki/Kompetisi_Sains_Nasional\" target=\"_blank\"\u003eWikipedia\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"discussion-of-ksn-2020-mathematics-questions\" class=\"headerLink\"\u003e\n \u003ca href=\"#discussion-of-ksn-2020-mathematics-questions\" class=\"header-mark\" aria-label=\"Direct link to section: Discussion of KSN 2020 Mathematics Questions\"\u003e\u003c/a\u003eDiscussion of KSN 2020 Mathematics Questions\u003c/h2\u003e\u003cp\u003eBelow we include a discussion of KSN questions in mathematics at district/city level in 2020\u003c/p\u003e\n\u003ciframe height=\"480\" src=\"https://drive.google.com/file/d/13ARnrpF4rptra2Hlv9Xohdy4hC_dZt-R/preview\" width=\"640\"\u003e\u003c/iframe\u003e", "language": "en" }, { "title" : "Responding to the Covid-19 Pandemic Test", "date_published" : "2020-09-17T14:29:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menyikapi-ujian-pandemi-covid-19/", "url" : "https://www.aantriono.com/en/menyikapi-ujian-pandemi-covid-19/", "summary": " IntroductionFriday sermon image First SermonAll praise is due to Allah, Lord of the worlds, who has bestowed upon us His guidance and mercy so that today we are gathered in this mosque to carry out His command — the Friday prayer and its rites. May blessings and peace be upon our role model, Prophet Muhammad SAW, who delivered the message of this religion.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikmo6rRRbCEBg2-b-MIMa-woG8rgPErdnBpSDVpEPCh-DBe6ja5iRmGlu2kp7k5TQmc4kCMxQM5iQ1JO325oyV-40x82ibpq73otA0LTVivwI7oaCtQntD7kTciQPrDcGFiSxv9ooIA54/w400-h188/khutbah.jpg\" title=\"Friday sermon\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikmo6rRRbCEBg2-b-MIMa-woG8rgPErdnBpSDVpEPCh-DBe6ja5iRmGlu2kp7k5TQmc4kCMxQM5iQ1JO325oyV-40x82ibpq73otA0LTVivwI7oaCtQntD7kTciQPrDcGFiSxv9ooIA54/w400-h188/khutbah.jpg\" data-sub-html=\"\u003ch2\u003eFriday sermon image\u003c/h2\u003e\u003cp\u003eFriday sermon\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikmo6rRRbCEBg2-b-MIMa-woG8rgPErdnBpSDVpEPCh-DBe6ja5iRmGlu2kp7k5TQmc4kCMxQM5iQ1JO325oyV-40x82ibpq73otA0LTVivwI7oaCtQntD7kTciQPrDcGFiSxv9ooIA54/w400-h188/khutbah.jpg' alt=\"Friday sermon\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eFriday sermon image\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch2 id=\"first-sermon\" class=\"headerLink\"\u003e\n \u003ca href=\"#first-sermon\" class=\"header-mark\" aria-label=\"Direct link to section: First Sermon\"\u003e\u003c/a\u003eFirst Sermon\u003c/h2\u003e\u003cp\u003eAll praise is due to Allah, Lord of the worlds, who has bestowed upon us His guidance and mercy so that today we are gathered in this mosque to carry out His command — the Friday prayer and its rites. May blessings and peace be upon our role model, Prophet Muhammad SAW, who delivered the message of this religion.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eO congregation blessed by Allah subhanahu wa ta\u0026rsquo;ala,\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eAllah has sworn to test humanity with five kinds of trials: fear, hunger, loss of wealth, death, and scarcity of fruits. As Allah says in Surah Al-Baqarah, verse 155:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eوَلَنَبْلُوَنَّكُمْ بِشَيْءٍ مِنَ الْخَوْفِ وَالْجُوعِ وَنَقْصٍ مِنَ الْأَمْوَالِ وَالْأَنْفُسِ وَالثَّمَرَاتِ ۗ وَبَشِّرِ الصَّابِرِينَ\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;And surely We shall test you with something of fear, hunger, loss of wealth, lives, and fruits. And give glad tidings to those who are patient.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eThis verse shows us that calamity is an inevitability every human being must face. Disasters, in whatever form, are in truth a manifestation of Allah\u0026rsquo;s mercy toward humanity. The various events that befall people are, in essence, tests of faith and behavior.\u003c/p\u003e\n\u003cp\u003eCurrently, humanity is being tested with the Covid-19 pandemic that originated in Wuhan, China. What should our attitude as Muslims be in facing this pandemic? There are several attitudes a Muslim should adopt:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFirst: Trust in Allah (\u003cem\u003etawakkul\u003c/em\u003e).\u003c/strong\u003e Every Muslim must firmly believe that everything that happens in this world is by His power and decree. We should surrender and place our trust in Allah. In Surah At-Taghabun, verse 11, Allah says:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eمَآ أَصَابَ مِن مُّصِيبَةٍ إِلَّا بِإِذْنِ ٱللَّهِ ۗ وَمَن يُؤْمِنۢ بِٱللَّهِ يَهْدِ قَلْبَهُۥ ۚ وَٱللَّهُ بِكُلِّ شَىْءٍ عَلِيمٌ\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;No calamity befalls except by Allah\u0026rsquo;s permission. And whoever believes in Allah — He will guide his heart. And Allah is All-Knowing of everything.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSecond: Guard the commands of Allah.\u003c/strong\u003e If we uphold Allah\u0026rsquo;s rules — carrying out His commands and avoiding His prohibitions — Allah will surely protect us. The Prophet SAW advised Ibn \u0026lsquo;Abbas:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eاحْفَظِ اللهَ يَحْفَظْكَ\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;Guard Allah, and Allah will guard you.\u0026rdquo; (HR. Tirmidhi)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThird: Remember the state of a believer between gratitude and patience.\u003c/strong\u003e The Prophet SAW said:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eعَجَبًا لأَمْرِ الْمُؤْمِنِ إِنَّ أَمْرَهُ كُلَّهُ خَيْرٌ\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;How wonderful is the affair of the believer! All of his affairs are good. If he experiences ease, he is grateful — and that is good for him. If he experiences hardship, he is patient — and that too is good for him.\u0026rdquo; (HR. Muslim)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFourth: Undertake effort and means (\u003cem\u003eikhtiar\u003c/em\u003e).\u003c/strong\u003e Seeking treatment does not contradict trust in Allah. To face an epidemic, the Prophet SAW taught:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eإِذَا سَمِعْتُمُ الطَّاعُونَ بِأَرْضٍ، فَلاَ تَدْخُلُوهَا، وَإِذَا وَقَعَ بِأرْضٍ، وَأَنْتُمْ فِيهَا، فَلاَ تَخْرُجُوا مِنْهَا\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;If you hear of a plague in a land, do not enter it. And if it breaks out in a land where you are, do not leave it.\u0026rdquo; (Muttafaqun \u0026lsquo;alaih)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFifth: Strengthen yourself with remembrance of Allah (\u003cem\u003edhikr\u003c/em\u003e).\u003c/strong\u003e The Prophet SAW said that whoever recites every morning and evening:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eبِسْمِ اللهِ الَّذِي لاَ يَضُرُّ مَعَ اسْمِهِ شَيْءٌ فِي الأَرْضِ وَلاَ فِي السَّمَاءِ وَهُوَ السَّمِيعُ العَلِيمُ\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003ethree times — nothing will harm him. (HR. Abu Daud and Tirmidhi)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSixth: Do not believe hoaxes.\u003c/strong\u003e A Muslim must verify news carefully. Allah says in Surah Al-Hujurat, verse 6:\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eيَٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُوٓا۟ إِن جَآءَكُمْ فَاسِقٌۢ بِنَبَإٍ فَتَبَيَّنُوٓا۟\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eMeaning: \u0026ldquo;O you who believe, if a wicked person comes to you with news, verify it, lest you harm a people in ignorance and become regretful over what you have done.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSeventh: Be patient.\u003c/strong\u003e Return to the verse from Al-Baqarah: \u0026ldquo;And give glad tidings to those who are patient.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eO congregation,\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eLet us continuously strive and pray that this Covid-19 pandemic ends soon, that Allah lifts this virus from us so that we may work and worship without anxiety. Aamiin.\u003c/p\u003e\n\u003ch2 id=\"second-sermon\" class=\"headerLink\"\u003e\n \u003ca href=\"#second-sermon\" class=\"header-mark\" aria-label=\"Direct link to section: Second Sermon\"\u003e\u003c/a\u003eSecond Sermon\u003c/h2\u003e\u003cp\u003eAll praise is due to Allah who has guided us to this, and we would not have been guided had Allah not guided us. May blessings and peace be upon Prophet Muhammad, his family, and his companions.\u003c/p\u003e\n\u003cp\u003eO Allah, forgive the believing men and women, the living among them and the deceased. Our Lord, grant us good in this world and good in the Hereafter, and protect us from the punishment of the Fire.\u003c/p\u003e\n\u003cp\u003eIndeed, Allah commands justice, good conduct, and giving to relatives, and forbids immorality, bad conduct, and oppression. He admonishes you that perhaps you will be reminded. So remember Allah; He will remember you. Be grateful to Him for His favors, and He will increase you. And the remembrance of Allah is greater.\u003c/p\u003e\n", "language": "en" }, { "title" : "Islamic New Year 1442 H", "date_published" : "2020-09-16T15:19:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pada-tahun-2020-m-ini-tahun-baru-islam/", "url" : "https://www.aantriono.com/en/pada-tahun-2020-m-ini-tahun-baru-islam/", "summary": " History and Meaning of the Islamic New Year 1442 HIslamic New Year 1442 H In 2020 AD, the Islamic New Year, 1 Muharram 1442 H, coincided with Thursday, August 8, 2020. Islam has 12 months in one year. The names of the months in the Hijri calendar are: Muharram, Safar, Rabiul Awal, Rabiul Akhir/Rabi\u0026rsquo;uts Tsany, Jumadil Awal, Jumadil Akhir/Jumadits Tsany, Rajab, Syakban, Ramadan, Syawal, Zulkaidah, and Zulhijah. Among the twelve months, there are what are called the sacred months (al-asyhur al-hurum): Zulkaidah, Zulhijah, Muharram, and Rajab.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"history-and-meaning-of-the-islamic-new-year-1442-h\" class=\"headerLink\"\u003e\n \u003ca href=\"#history-and-meaning-of-the-islamic-new-year-1442-h\" class=\"header-mark\" aria-label=\"Direct link to section: History and Meaning of the Islamic New Year 1442 H\"\u003e\u003c/a\u003eHistory and Meaning of the Islamic New Year 1442 H\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXfCaMlBjOs_lpHuVHuJXX9_gBW0mvvuBQYUogq6wfycS2Ctl4ryYmurm4xRIgnVH2AUAn2xAOF9U9e7l7ORw_IiKMgs0M_F-2tQUmx1TujBSqPNId90uy7uw3fmBK8vv10pwuZcTdmAc/w400-h224/tahun\u0026#43;baru\u0026#43;hijriyah.jpg\" title=\"Islamic New Year 1442 H\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXfCaMlBjOs_lpHuVHuJXX9_gBW0mvvuBQYUogq6wfycS2Ctl4ryYmurm4xRIgnVH2AUAn2xAOF9U9e7l7ORw_IiKMgs0M_F-2tQUmx1TujBSqPNId90uy7uw3fmBK8vv10pwuZcTdmAc/w400-h224/tahun\u0026#43;baru\u0026#43;hijriyah.jpg\" data-sub-html=\"\u003ch2\u003eIslamic New Year 1442 H\u003c/h2\u003e\u003cp\u003eIslamic New Year 1442 H\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXfCaMlBjOs_lpHuVHuJXX9_gBW0mvvuBQYUogq6wfycS2Ctl4ryYmurm4xRIgnVH2AUAn2xAOF9U9e7l7ORw_IiKMgs0M_F-2tQUmx1TujBSqPNId90uy7uw3fmBK8vv10pwuZcTdmAc/w400-h224/tahun+baru+hijriyah.jpg' alt=\"Islamic New Year 1442 H\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eIslamic New Year 1442 H\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eIn 2020 AD, the Islamic New Year, 1 Muharram 1442 H, coincided with Thursday, August 8, 2020. Islam has 12 months in one year. The names of the months in the Hijri calendar are: Muharram, Safar, Rabiul Awal, Rabiul Akhir/Rabi\u0026rsquo;uts Tsany, Jumadil Awal, Jumadil Akhir/Jumadits Tsany, Rajab, Syakban, Ramadan, Syawal, Zulkaidah, and Zulhijah. Among the twelve months, there are what are called the sacred months (\u003cem\u003eal-asyhur al-hurum\u003c/em\u003e): Zulkaidah, Zulhijah, Muharram, and Rajab.\u003c/p\u003e\n\u003ch2 id=\"origin-of-the-hijri-calendar\" class=\"headerLink\"\u003e\n \u003ca href=\"#origin-of-the-hijri-calendar\" class=\"header-mark\" aria-label=\"Direct link to section: Origin of the Hijri Calendar\"\u003e\u003c/a\u003eOrigin of the Hijri Calendar\u003c/h2\u003e\u003cp\u003eThe Hijri New Year, or Islamic New Year, is an important day for Muslims because it marks a significant event in Islamic history: the migration (\u003cem\u003ehijrah\u003c/em\u003e) of the Prophet Muhammad SAW from Mecca to Medina in 622 CE. This historic event occurred on 1 Muharram, the first day of the Hijri calendar.\u003c/p\u003e\n\u003cp\u003eThe Hijri calendar was not officially established during the time of the Prophet SAW. It was only instituted during the era of the second Rightly-Guided Caliph, Umar al-Faruq RA. There were several suggestions from the Companions for determining the starting date: some proposed that the Islamic year begin with the birth of the Prophet Muhammad SAW, others suggested the day he was appointed as a prophet and messenger. However, the view that prevailed was to begin the Islamic calendar from the date of the Prophet\u0026rsquo;s migration.\u003c/p\u003e\n\u003cp\u003eThis decision was made to commemorate the importance of the Hijrah as a paradigm shift in Islamic history. It was the first time a prophet and messenger established governance amid great difficulty, built diplomatic relations with other regions, and delivered the message of Islam globally — enabling Islam to spread throughout the world.\u003c/p\u003e\n\u003ch2 id=\"three-key-meanings-of-the-islamic-new-year\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-key-meanings-of-the-islamic-new-year\" class=\"header-mark\" aria-label=\"Direct link to section: Three Key Meanings of the Islamic New Year\"\u003e\u003c/a\u003eThree Key Meanings of the Islamic New Year\u003c/h2\u003e\u003cp\u003eFor Muslims, the Islamic New Year holds deep and special meaning. Here are three meanings of the Islamic New Year:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eThe migration of the Prophet Muhammad SAW.\u003c/strong\u003e 1 Muharram is also commemorated as a reminder of the momentous event when the Prophet migrated from Mecca to Medina. After the migration, Islam experienced rapid growth and spread increasingly to Mecca and surrounding regions.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eThe struggle of the Prophet and his Companions.\u003c/strong\u003e The Islamic New Year is also understood as the embodiment of an unwavering spirit of struggle in spreading Islam by the Prophet Muhammad SAW and his Companions. Despite many challenges and obstacles, they never gave up or became pessimistic. The Prophet even left his birthplace, relatives, and possessions solely to fulfill the command and revelation given by Allah SWT.\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003e\u003cstrong\u003eSelf-reflection.\u003c/strong\u003e The turn of the Islamic New Year is also a moment for self-reflection (\u003cem\u003emuhasabah\u003c/em\u003e). As time continues to pass, the arrival of the Islamic New Year encourages Muslims to be more introspective, to reflect upon all actions and deeds carried out over the past twelve months, and to consider what needs to be improved in the year ahead.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eMay the Islamic New Year 1442 H bring a renewed spirit — a spirit of change, of migrating from what is harmful toward what is good, and of remaining steadfast (\u003cem\u003eistiqamah\u003c/em\u003e) upon the path of goodness and truth, which is \u003cem\u003eDinul Islam\u003c/em\u003e. Aamiin.\u003c/p\u003e", "language": "en" }, { "title" : "Syllabus", "date_published" : "2020-09-16T14:07:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/silabus/", "url" : "https://www.aantriono.com/en/silabus/", "summary": " Introduction SyllabusA syllabus is a set of plans and arrangements for the implementation of learning and assessment that are arranged systematically, containing interrelated components to achieve mastery of basic competencies. Syllabus development steps (Trianto, 2010: 99):\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"\" class=\"headerLink\"\u003e\n \u003ca href=\"#\" class=\"header-mark\" aria-label=\"Direct link to section: \"\u003e\u003c/a\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtp4XZzE3C_4bzCzBgxkFLNfFWuYjcsr1NlDYxOz1tp8AyFk0vQK2mYcA1qMo6EWxY4xkzSShUWtj2zldlJjnxul-AawEaoADrVqNA7rIVC92iNRhl30A5S4MLdWoorU-6O4iFvjnG29Y/s254/silabus.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"silabus mata pelajaran matematika\" border=\"0\" data-original-height=\"198\" data-original-width=\"254\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtp4XZzE3C_4bzCzBgxkFLNfFWuYjcsr1NlDYxOz1tp8AyFk0vQK2mYcA1qMo6EWxY4xkzSShUWtj2zldlJjnxul-AawEaoADrVqNA7rIVC92iNRhl30A5S4MLdWoorU-6O4iFvjnG29Y/s16000/silabus.jpg\" title=\"Gambar silabus\" /\u003e\u003c/a\u003e\u003c/h2\u003e\u003ch2 id=\"syllabus\" class=\"headerLink\"\u003e\n \u003ca href=\"#syllabus\" class=\"header-mark\" aria-label=\"Direct link to section: Syllabus\"\u003e\u003c/a\u003eSyllabus\u003c/h2\u003e\u003cp\u003eA syllabus is a set of plans and arrangements for the implementation of learning and assessment that are arranged systematically, containing interrelated components to achieve mastery of basic competencies.\n\u003cstrong\u003eSyllabus development steps (Trianto, 2010: 99):\u003c/strong\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eReviewing Competency Standards and Basic Competencies. Review the SK and KD of subjects as stated in the Content Standards.\u003c/li\u003e\n\u003cli\u003eIdentifying Main/Learning Material. Identify the main/learning material that supports the achievement of KD.\u003c/li\u003e\n\u003cli\u003eDeveloping Learning Activities. Learning activities are designed to provide learning experiences that involve mental and physical processes in order to achieve KD.\u003c/li\u003e\n\u003cli\u003eFormulating Competency Achievement Indicators. Indicators are markers of KD achievement. Indicators are used as a basis for developing assessment tools.\u003c/li\u003e\n\u003cli\u003eDetermining the Type of Assessment. Assessment of students' achievement of basic competencies is carried out based on indicators. Assessment is carried out using tests in written form.\u003c/li\u003e\n\u003cli\u003eDetermining Time Allocation. Determination of time allocation for each KD is based on the effective number of weeks and subject time allocation per week. Time allocation is an estimate of the average time to master the KD required by various students.\u003c/li\u003e\n\u003cli\u003eDetermining Learning Resources. Determination of learning resources is based on SK and KD as well as main/learning material, learning activities, and indicators of competency achievement.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"syllabus-benefits\" class=\"headerLink\"\u003e\n \u003ca href=\"#syllabus-benefits\" class=\"header-mark\" aria-label=\"Direct link to section: Syllabus Benefits\"\u003e\u003c/a\u003eSyllabus Benefits\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eThe syllabus is useful as a guide for further development of learning tools, starting from planning, managing learning activities and developing assessments.\u003c/li\u003e\n\u003cli\u003eThe syllabus is useful as a guide in further learning development, such as creating learning plans, managing learning activities and developing assessment systems. The syllabus is the main source in preparing learning plans, the structure of learning plans for one Competency Standard and one Basic Competency.\u003c/li\u003e\n\u003cli\u003eThe syllabus is also useful as a guide for planning the management of learning activities, for example classical, small group or individual learning activities. Likewise, a syllabus is very useful for developing an assessment system.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"fill-in-the-syllabus\" class=\"headerLink\"\u003e\n \u003ca href=\"#fill-in-the-syllabus\" class=\"header-mark\" aria-label=\"Direct link to section: Fill in the Syllabus\"\u003e\u003c/a\u003eFill in the Syllabus\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eSubject identity\u003c/li\u003e\n\u003cli\u003eSchool identity includes the name of the educational unit and class\u003c/li\u003e\n\u003cli\u003ecore competency,\u003c/li\u003e\n\u003cli\u003eBasic competencies\u003c/li\u003e\n\u003cli\u003eTheme (specifically SD/MI/SDLB/Package A/etc)\u003c/li\u003e\n\u003cli\u003eThe main material contains relevant facts, concepts, principles and procedures, and is written in the form of bullet points in accordance with the formulation of competency achievement indicators;\u003c/li\u003e\n\u003cli\u003eLearning, namely activities carried out by educators and students to achieve the expected competencies;\u003c/li\u003e\n\u003cli\u003eAssessment is the process of collecting and processing information to determine the achievement of learning outcomes\u003c/li\u003e\n\u003cli\u003eTime Allocation\u003c/li\u003e\n\u003cli\u003eLearning resources can be books, print and electronic media, the natural environment or other relevant learning resources.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"syllabus-development-principles\" class=\"headerLink\"\u003e\n \u003ca href=\"#syllabus-development-principles\" class=\"header-mark\" aria-label=\"Direct link to section: Syllabus Development Principles\"\u003e\u003c/a\u003eSyllabus Development Principles\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eScientific; All material and activities contained in the syllabus must be correct and scientifically accountable.\u003c/li\u003e\n\u003cli\u003eRelevant; The scope, depth, level of difficulty and sequence of presentation of material in the syllabus are appropriate to the level of physical, intellectual, social, emotional and spiritual development of students.\u003c/li\u003e\n\u003cli\u003eSystematic; Syllabus components are functionally interconnected in achieving competency.\u003c/li\u003e\n\u003cli\u003eConsistency; There is a consistent relationship between basic competencies, indicators, main material, learning experiences, learning resources and assessment systems.\u003c/li\u003e\n\u003cli\u003eAdequacy; The coverage of indicators, main material, learning experiences, learning resources and assessment systems is sufficient to support the achievement of basic competencies.\u003c/li\u003e\n\u003cli\u003eActual \u0026 Contextual; The coverage of indicators, subject matter, learning experiences, learning resources and assessment systems takes into account the latest developments in science, technology and art in real life, and events that occur.\u003c/li\u003e\n\u003cli\u003eFlexible; All syllabus components can accommodate the diversity of students, educators, as well as the dynamics of change that occur in schools and societal demands.\u003c/li\u003e\n\u003cli\u003eComprehensive; Syllabus components cover the entire competency domain (Cognitive, affective, psychomotor) or according to the essence of each subject.\u003c/li\u003e\n\u003c/ol\u003e\nBelow I have included an example of a mathematics syllabus for class VIII SMP curriculum 2013\n\u003ciframe height=\"480\" src=\"https://drive.google.com/file/d/1NN90-m-BaZah3_RsPmSRxDiQKP7-MGK8/preview\" width=\"640\"\u003e\u003c/iframe\u003e\n", "language": "en" }, { "title" : "Designing and Creating Google Classroom", "date_published" : "2020-09-16T13:18:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/merancang-dan-membuat-google-classroom/", "url" : "https://www.aantriono.com/en/merancang-dan-membuat-google-classroom/", "summary": " Introduction Create an Online Learning Class Using Google Classroom Below are the steps for creating an online learning class by using Google Classroom. For MStart creating a new class in Google Classroom by going to the page address **classroom.google.com . **Please log in with using your Gmail account. If you have logged in to gmail, you can also log in to Google Classroom, by clicking on the Classroom option on the Gmail account menu. ", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003ch2 id=\"create-an-online-learning-class-using-google-classroom\" class=\"headerLink\"\u003e\n \u003ca href=\"#create-an-online-learning-class-using-google-classroom\" class=\"header-mark\" aria-label=\"Direct link to section: Create an Online Learning Class Using Google Classroom\"\u003e\u003c/a\u003eCreate an Online Learning Class Using Google Classroom\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQ2ZQLom3pavEys-MG1N71gXrYs3DcCmSkizEAlzPRqk8vpO622QhEFXMEQVkx3RjJv5TVKMCWpKGMTEozCrXRe1u5b53th-CGosIlsK_ZFRQDFpfpkk_w_esRUg3FfbHCchq8qsn1pRs/s1600/GC.jpeg\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"675\" data-original-width=\"1200\" height=\"111\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQ2ZQLom3pavEys-MG1N71gXrYs3DcCmSkizEAlzPRqk8vpO622QhEFXMEQVkx3RjJv5TVKMCWpKGMTEozCrXRe1u5b53th-CGosIlsK_ZFRQDFpfpkk_w_esRUg3FfbHCchq8qsn1pRs/s200/GC.jpeg\" width=\"200\" /\u003e\u003c/a\u003e\nBelow are the steps for creating an online learning class by using Google Classroom. For MStart creating a new class in Google Classroom by going to the page address **\u003ca href=\"http://classroom.google.com/\" style=\"background-color: white;\"\u003eclassroom.google.com\u003c/a\u003e\n. **Please log in with using your Gmail account. If you have logged in to gmail, you can also log in to Google Classroom, by clicking on the Classroom option on the Gmail account menu.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjExUyE8ImP7j3ZvYO9KunHLzuok7r-lMGBsT14pPe5C9AXs2Kt8LuW2k1JGn3hdI9WBdZsI0-i6QNYDhrFffuhdM23FzZ1vpnZEKOSoJGVeZHS-amKVMOXO3xNM0rKErA11M8ph-5IHeQ/s1600/Screenshot_6.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"470\" data-original-width=\"1313\" height=\"114\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjExUyE8ImP7j3ZvYO9KunHLzuok7r-lMGBsT14pPe5C9AXs2Kt8LuW2k1JGn3hdI9WBdZsI0-i6QNYDhrFffuhdM23FzZ1vpnZEKOSoJGVeZHS-amKVMOXO3xNM0rKErA11M8ph-5IHeQ/s320/Screenshot_6.png\" width=\"320\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eAfter clicking the 9 dot icon, the following Google applications will appear, then select and click the Classroom icon\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEyYgQUUrJdRhCBdP6yxyyxvqvBvm955jmcuZCJeMH84ndsaewRMOYMcdK-Dd-hxJqItqwlYRD3aSkS6q6pop9FCjSfIEyQtLYbLUwCgnLO8rcZtA34ro2HzXDtvg1V4JvShLqz5dt3Jo/s1600/Screenshot_7.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"313\" data-original-width=\"178\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEyYgQUUrJdRhCBdP6yxyyxvqvBvm955jmcuZCJeMH84ndsaewRMOYMcdK-Dd-hxJqItqwlYRD3aSkS6q6pop9FCjSfIEyQtLYbLUwCgnLO8rcZtA34ro2HzXDtvg1V4JvShLqz5dt3Jo/s200/Screenshot_7.png\" width=\"113\" /\u003e\u003c/a\u003e\nTo create a class, click the plus sign (+) icon and select create class (create class).\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0fN7_DRE_HYOpIgSCNnG7EEP-pWsgHycYiZQdcYumeXUcQ8mU-Ame5fr5psj6hLkJkejmV93iWwNJQlY0vMRAX4rxQrEDXTU_eBk8LFJud0EebKPIERv3gWBfr39n1bP7tsdwrKdIgZ8/s1600/Screenshot_8.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"405\" data-original-width=\"1281\" height=\"101\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0fN7_DRE_HYOpIgSCNnG7EEP-pWsgHycYiZQdcYumeXUcQ8mU-Ame5fr5psj6hLkJkejmV93iWwNJQlY0vMRAX4rxQrEDXTU_eBk8LFJud0EebKPIERv3gWBfr39n1bP7tsdwrKdIgZ8/s320/Screenshot_8.png\" width=\"320\" /\u003e\u003c/a\u003e\nWhen starting to create a class, there will be an approval warning. The approval state that you do not use Google Classroom for school with students, for example a class that you create just to study together. You must use G Suite for Education (Google for Education) if you wish using Google Classroom for schools and students. Next, the following text will appear and check the small box at the bottom, then click continue\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjv6EI0HQ1Kn4JL5Df8Jdz9_VKPsl6pl5CieMbjnOKk5PQfV-pzGS9PSqLU4ltITU9SetoqQ1Nux3jr2rUOjYISbu-Zv4Kll6J1QQwge3kp5ycWDS0EQIzILyj2THyoyfQzoFSRfRPfWuI/s1600/Screenshot_5.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"528\" data-original-width=\"345\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjv6EI0HQ1Kn4JL5Df8Jdz9_VKPsl6pl5CieMbjnOKk5PQfV-pzGS9PSqLU4ltITU9SetoqQ1Nux3jr2rUOjYISbu-Zv4Kll6J1QQwge3kp5ycWDS0EQIzILyj2THyoyfQzoFSRfRPfWuI/s200/Screenshot_5.png\" width=\"130\" /\u003e\u003c/a\u003e\nAfter clicking continue, you will go to the next stage, namely filling in the initial form class opening.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwIowgQmo9Nq5R2iE7oE3qibhbJVnw4o6QV12nwn0PDhQCP9MFD8lPkausW7BWa5WOdfNTuSL2P7JoFpfzyDcq6-AE1EzKZNnEeV8IFrQiVxKuUdqRVLkQXEafnoAD3DgT6FzXZbURH4U/s1600/Screenshot_9.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"409\" data-original-width=\"558\" height=\"232\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwIowgQmo9Nq5R2iE7oE3qibhbJVnw4o6QV12nwn0PDhQCP9MFD8lPkausW7BWa5WOdfNTuSL2P7JoFpfzyDcq6-AE1EzKZNnEeV8IFrQiVxKuUdqRVLkQXEafnoAD3DgT6FzXZbURH4U/s320/Screenshot_9.png\" width=\"320\" /\u003e\u003c/a\u003e\nThe class name must be filled in, in the section menu you can fill in a brief description, class level and class schedule. Click create, your learning class in Google Classroom is ready to be filled.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaTWEUHoxo6Ns6LRa_mJspwUX6fSKc7uaxIhB4m72rgYnoiz_JMUkZ5hdhNmWAghWbj7diZ4RBfUEHmhFGx0YGtsQhdx6RfUZrXbCNkZDg4TumanMvWrEG9nqBBv1VYFJLjrkna-oarMc/s1600/Screenshot_10.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"381\" data-original-width=\"767\" height=\"157\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaTWEUHoxo6Ns6LRa_mJspwUX6fSKc7uaxIhB4m72rgYnoiz_JMUkZ5hdhNmWAghWbj7diZ4RBfUEHmhFGx0YGtsQhdx6RfUZrXbCNkZDg4TumanMvWrEG9nqBBv1VYFJLjrkna-oarMc/s320/Screenshot_10.png\" width=\"320\" /\u003e\u003c/a\u003e\nIn the initial display of the class that you have created, the class name, short description and class code will appear. The menus that appear are the Forum Menu, Class Assignments, Members, and Grades. The class code that we will share with teachers or students is on the right under the class description, while the class theme menu is in the menu on the right. To change the theme we can select and click \u0026ldquo;Select theme\u0026rdquo;, and images will appear as follows:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi16intgeK36wFdpbEbVdd9lATtxB-zUVMr2jQAXWQcdrecCzaacy47PaCe_tJNMLbXBIrd5M_WpldC7hcZMOFRxEK_idohgKrbk2yPnmEQ5KW_tgZA3Qb7-anx46dktq7HiB4fBsH37iI/s1600/Screenshot_11.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"499\" data-original-width=\"808\" height=\"196\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi16intgeK36wFdpbEbVdd9lATtxB-zUVMr2jQAXWQcdrecCzaacy47PaCe_tJNMLbXBIrd5M_WpldC7hcZMOFRxEK_idohgKrbk2yPnmEQ5KW_tgZA3Qb7-anx46dktq7HiB4fBsH37iI/s320/Screenshot_11.png\" width=\"320\" /\u003e\u003c/a\u003e\nIt is best to choose a class theme according to the subject being taught and then click select class theme.\nThat\u0026rsquo;s how to create an online learning class using Google Classroom services. Once the class is ready, the next step is to invite the teacher, students and prepare the activities to be carried out such as attendance list, materials, discussions, assignments, assessments and so on. All will be discussed in the next tutorial.\u003c/p\u003e", "language": "en" }, { "title" : "Two Songs That Soothe the Heart", "date_published" : "2020-09-14T14:23:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/dua-kidung-penawar-hati/", "url" : "https://www.aantriono.com/en/dua-kidung-penawar-hati/", "summary": " Introduction to Devotional PrayersA person praying Here are two devotional verses that can soothe the heart.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction-to-devotional-prayers\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction-to-devotional-prayers\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction to Devotional Prayers\"\u003e\u003c/a\u003eIntroduction to Devotional Prayers\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_MXspketZ3ZZOxAOcG4N8KfS8u7MDdZxdLP8_6wjIYsGbvz-pfIgyQiXGV-6IY6quVM-u3kP-nAIiPTwNUQMHjw0H_NBu6MGRg8yTuMYTBETHcCp8ADQMzP792t997x8g2jXCzi5UKHQ/w200-h134/15444670435c0eb263890bb.jpg\" title=\"Praying to Allah\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_MXspketZ3ZZOxAOcG4N8KfS8u7MDdZxdLP8_6wjIYsGbvz-pfIgyQiXGV-6IY6quVM-u3kP-nAIiPTwNUQMHjw0H_NBu6MGRg8yTuMYTBETHcCp8ADQMzP792t997x8g2jXCzi5UKHQ/w200-h134/15444670435c0eb263890bb.jpg\" data-sub-html=\"\u003ch2\u003eA person praying\u003c/h2\u003e\u003cp\u003ePraying to Allah\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_MXspketZ3ZZOxAOcG4N8KfS8u7MDdZxdLP8_6wjIYsGbvz-pfIgyQiXGV-6IY6quVM-u3kP-nAIiPTwNUQMHjw0H_NBu6MGRg8yTuMYTBETHcCp8ADQMzP792t997x8g2jXCzi5UKHQ/w200-h134/15444670435c0eb263890bb.jpg' alt=\"Praying to Allah\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eA person praying\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eHere are two devotional verses that can soothe the heart.\u003c/p\u003e\n\u003ch2 id=\"first-devotion-longing-for-goodness\" class=\"headerLink\"\u003e\n \u003ca href=\"#first-devotion-longing-for-goodness\" class=\"header-mark\" aria-label=\"Direct link to section: First Devotion: Longing for Goodness\"\u003e\u003c/a\u003eFirst Devotion: Longing for Goodness\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eO Allah, I long to do what is good, yet my strength is so limited.\u003c/li\u003e\n\u003cli\u003eForgive my shortcomings, and do not leave me to the burden of my sins.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"second-devotion-the-al-itiraaf-poem\" class=\"headerLink\"\u003e\n \u003ca href=\"#second-devotion-the-al-itiraaf-poem\" class=\"header-mark\" aria-label=\"Direct link to section: Second Devotion: The Al-I\u0026rsquo;tiraaf Poem\"\u003e\u003c/a\u003eSecond Devotion: The Al-I\u0026rsquo;tiraaf Poem\u003c/h2\u003e\u003cblockquote\u003e\n \u003cp\u003e\u003cspan lang=\"ar\"\u003eإلهي لست للفردوس أهلاً\u003c/span\u003e\n\u003cspan lang=\"ar\"\u003eولا أقوى على نار الجحيم\u003c/span\u003e\n\u003cspan lang=\"ar\"\u003eفهب لي توبة واغفر ذنوبي\u003c/span\u003e\n\u003cspan lang=\"ar\"\u003eفإنك غافر الذنب العظيم\u003c/span\u003e\n\u003cspan lang=\"ar\"\u003eوعاملني معاملة الكريم\u003c/span\u003e\n\u003cspan lang=\"ar\"\u003eوثبتني على النهج القويم\u003c/span\u003e\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003e\u0026ldquo;O my Lord, I am not worthy of Paradise, yet I am not strong enough to endure the fire of Hell. Grant me repentance and forgive my sins, for You are the One who forgives even the gravest sin. Treat me with the generosity that belongs to You, and keep me firm upon the straight path.\u0026rdquo;\u003c/p\u003e", "language": "en" }, { "title" : "Overview of eLearner", "date_published" : "2020-09-12T13:36:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sekilas-tentang-epelajar/", "url" : "https://www.aantriono.com/en/sekilas-tentang-epelajar/", "summary": " Introduction During the Covid-19 pandemic, there are many online learning applications that schools can use to carry out online learning, such as Edmodo, Epelajar, Google Classroom, Moodle and so on. Of course, each application has its own advantages and disadvantages. For example, some of the advantages of the Epelajar application are:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxxJJGYl-W90mAaq43rtOxb8YuOF8rOvFlxpKU2S6_ifGea6BSWzVcYHhBkLFrNdbkzX-ig1U7DhwxmRhmnZ9aH7-lJt9Jmc3Wz1zheRAogTb70uyiRJYiQQB9ovuK8O3FKkKzfj3lCYs/s425/epelajar+judul.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"pembelajaran online menggunakan aplikasi Epelajar\" border=\"0\" data-original-height=\"425\" data-original-width=\"341\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxxJJGYl-W90mAaq43rtOxb8YuOF8rOvFlxpKU2S6_ifGea6BSWzVcYHhBkLFrNdbkzX-ig1U7DhwxmRhmnZ9aH7-lJt9Jmc3Wz1zheRAogTb70uyiRJYiQQB9ovuK8O3FKkKzfj3lCYs/w161-h200/epelajar+judul.png\" title=\"Gambar aplikasi Epelajar\" width=\"161\" /\u003e\u003c/a\u003e\nDuring the Covid-19 pandemic, there are many online learning applications that schools can use to carry out online learning, such as Edmodo, Epelajar, Google Classroom, Moodle and so on. Of course, each application has its own advantages and disadvantages.\nFor example, some of the advantages of the Epelajar application are:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eSystem/Technical:\u003c/li\u003e\n\u003c/ol\u003e\n\u003cul\u003e\n\u003cli\u003eThe only e-learning based conference platform\u003c/li\u003e\n\u003cli\u003eExcellent audio and video quality\u003c/li\u003e\n\u003cli\u003eReal time chat feature that can be monitored by the teacher\u003c/li\u003e\n\u003cli\u003eLighter bandwidth due to video streaming from one frame\u003c/li\u003e\n\u003c/ul\u003e\n2. User Usage:\n\u003cul\u003e\n\u003cli\u003eUser Friendly\u003c/li\u003e\n\u003cli\u003eSecurity and privacy are protected\u003c/li\u003e\n\u003cli\u003eSchools can monitor teachers regarding the lesson plans taught on that day\u003c/li\u003e\n\u003cli\u003eSchedules and rooms can be adjusted\u003c/li\u003e\n\u003cli\u003ePresence status is automatically recorded\u003c/li\u003e\n\u003cli\u003eFull access by the teacher such as activating/deactivating student video and audio tools, share screen, real time chat, question and answer and live presentations.\u003c/li\u003e\n\u003c/ul\u003e\nTo use the Epelajar application, schools must register first to obtain an account which is then used to access the application. If the school already has an account, you can access the Epelajar application by creating an accountopen your browser and type *live.epelajar.id/school/login*\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKeVgANxho_o7UMVaVQL0slP8ZObyBZ3jmiSFptHV_LfkOqchArzyB-ekTVXPPwUD_8mJmq4qGiRtIaO9jvjtKuAvd3LIMw6nkiLRu-MxjvIUnNAFK9Agj_59_M6lXnEtw3n6k5KDLbOs/s737/2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"543\" data-original-width=\"737\" height=\"148\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKeVgANxho_o7UMVaVQL0slP8ZObyBZ3jmiSFptHV_LfkOqchArzyB-ekTVXPPwUD_8mJmq4qGiRtIaO9jvjtKuAvd3LIMw6nkiLRu-MxjvIUnNAFK9Agj_59_M6lXnEtw3n6k5KDLbOs/w200-h148/2.png\" width=\"200\" /\u003e\u003c/a\u003e\nFill in your registered school email and password then click login. After successful login it will be visible The menu displays in the Epelajar application are:\n## 1. Home\nit looks like in the following picture:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoeqEvMKB9NwNfQ5DL2vQi_xciyhM9wJ_KbIdmD_1rFl-hjcwJWXrF9BovxUTWGKgK2a3Fom8Ppje4oJvArNjU9exwTFQVk0XCKJyutcdnFSm3XdKqZKUFm8bbHY__zZwrHfOGLiIGuik/s1139/tampilan+fitur+home.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"418\" data-original-width=\"1139\" height=\"73\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoeqEvMKB9NwNfQ5DL2vQi_xciyhM9wJ_KbIdmD_1rFl-hjcwJWXrF9BovxUTWGKgK2a3Fom8Ppje4oJvArNjU9exwTFQVk0XCKJyutcdnFSm3XdKqZKUFm8bbHY__zZwrHfOGLiIGuik/w200-h73/tampilan+fitur+home.png\" width=\"200\" /\u003e\u003c/a\u003e\nIt contains the school name, notification icon, class, subjects, teachers and students.\n## 2. Class\nit looks like the following image:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTtGkZfRQFQP_GjkFzlyNT6p_MvvRngw5XA4caiPunxFbqZTyWpxzhDkbuOe7mqg_DZ6PtcQDSmZghsVCDesG41F8Zno3cIjPmkmztV_XvA5bdXAFs_001wHDNmmNdXJdH-Gaayrc1XbA/s200/kelas1.png\"\u003e\u003cimg border=\"0\" data-original-height=\"427\" data-original-width=\"1159\" height=\"73\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTtGkZfRQFQP_GjkFzlyNT6p_MvvRngw5XA4caiPunxFbqZTyWpxzhDkbuOe7mqg_DZ6PtcQDSmZghsVCDesG41F8Zno3cIjPmkmztV_XvA5bdXAFs_001wHDNmmNdXJdH-Gaayrc1XbA/s200/kelas1.png\" width=\"200\" /\u003e\u003c/a\u003e\nIn the class menu there are features for number of views, search, import, add, edit and delete.\n## 3. Subjects\nit looks like the following image:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiag2aOy3UyHrFq90-a3jjVkBb6mOG_aX8HViS5b9c1-z0-QIpI-UCaF0IZ1X9uf1qoew8vUP82aBkfp7jzct-4eRLlUFMPyUFq9BXBtxnX06U-vZkw3OZvWygifBU31XDoJ5t9CUEkURU/s200/mapel.png\"\u003e\u003cimg border=\"0\" data-original-height=\"414\" data-original-width=\"1156\" height=\"71\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiag2aOy3UyHrFq90-a3jjVkBb6mOG_aX8HViS5b9c1-z0-QIpI-UCaF0IZ1X9uf1qoew8vUP82aBkfp7jzct-4eRLlUFMPyUFq9BXBtxnX06U-vZkw3OZvWygifBU31XDoJ5t9CUEkURU/s200/mapel.png\" width=\"200\" /\u003e\u003c/a\u003e\nIt contains features for viewing, searching, importing, adding, editing and deleting.\n## 4. Schedule\nit looks like the following picture:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXa-hwgegH2E3sKEaQqFMTx58s4OisxiYYgF8MeWzrEPwpvOuxybc2jgqiQ0ArS8t7GprzlsKRyA2KhMQh70VP6XTgjC0pgjNfOypIBK-jwuxs6nO7lVjHuEKN6DRi3sDOe1L8xgnRCjw/s1154/tampilan+jadwal.png\" style=\"font-family: \"times new roman\"; margin-left: 1em; margin-right: 1em; text-align: center;\"\u003e\u003cimg border=\"0\" data-original-height=\"353\" data-original-width=\"1154\" height=\"61\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXa-hwgegH2E3sKEaQqFMTx58s4OisxiYYgF8MeWzrEPwpvOuxybc2jgqiQ0ArS8t7GprzlsKRyA2KhMQh70VP6XTgjC0pgjNfOypIBK-jwuxs6nO7lVjHuEKN6DRi3sDOe1L8xgnRCjw/w200-h61/tampilan+jadwal.png\" width=\"200\" /\u003e\u003c/a\u003e\nIt contains features for number of subjects, search, create, details, edit and delete.\n## 5. Teacher\nhow it looks like the following picture:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrJUSX0tHd4LLlI4WUw97PRMKVR1LanC71fh4ld1o8X5xeVHJunficWePI-7C7mk0r0EnQPMNsPaeEMzXdRIz517YEsI8llv5IjsiYNmy8ZCqxYKXFt4AO2pkJ9ixY2FWCFiZVt6984JE/s1163/tampilan+fitur+guru.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"418\" data-original-width=\"1163\" height=\"72\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrJUSX0tHd4LLlI4WUw97PRMKVR1LanC71fh4ld1o8X5xeVHJunficWePI-7C7mk0r0EnQPMNsPaeEMzXdRIz517YEsI8llv5IjsiYNmy8ZCqxYKXFt4AO2pkJ9ixY2FWCFiZVt6984JE/w200-h72/tampilan+fitur+guru.png\" width=\"200\" /\u003e\u003c/a\u003e\nIn it there are features for viewing, searching, importing, adding, editing and deleting.\n## 6. Students\nit looks like the following image:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz2fFu8vYInSkNwpUDiDemVOi5FWaQo42qBVW1xo2rq2GjNFRKR8uNOVModFlBKFOX-ulHXsyXQe6WTIAlJ2r12maTbmJF9gWzF_WEi0jHIj4xNoUdD5fvFy0Jw65EIrVcVm-_UL428fY/s1160/tampilan+fitur+siswa.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"431\" data-original-width=\"1160\" height=\"74\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiz2fFu8vYInSkNwpUDiDemVOi5FWaQo42qBVW1xo2rq2GjNFRKR8uNOVModFlBKFOX-ulHXsyXQe6WTIAlJ2r12maTbmJF9gWzF_WEi0jHIj4xNoUdD5fvFy0Jw65EIrVcVm-_UL428fY/w200-h74/tampilan+fitur+siswa.png\" width=\"200\" /\u003e\u003c/a\u003e\nIn it there are features for number of views, classes, search, import, add, details, edit and delete.\n**\n\u003col start=\"7\"\u003e\n\u003cli\u003eRPP Report,** it looks like the following picture:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvin4Z5G0iR9jE6Xb915_KiI75M5WbYgT2FhvaMeJKSDmWq9aN25LtAjCucIgL186_qPD716JQVZYTZAQ9dFyLEtsOX3liaCCQ32Tb3B-cj3t_Uue0R9ChTQgiDkK7gP_QmcQMJop7fiw/s1175/fitur+report+rpp.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"310\" data-original-width=\"1175\" height=\"53\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvin4Z5G0iR9jE6Xb915_KiI75M5WbYgT2FhvaMeJKSDmWq9aN25LtAjCucIgL186_qPD716JQVZYTZAQ9dFyLEtsOX3liaCCQ32Tb3B-cj3t_Uue0R9ChTQgiDkK7gP_QmcQMJop7fiw/w200-h53/fitur+report+rpp.png\" width=\"200\" /\u003e\u003c/a\u003e\nIn it there are features for the number of views, search and show options. If the show option is opened, it will look like the following image:\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6mQBB8l43_RfvbWygBnf7uTikJJnSTuKD68qTFW9FK4Qphb5_sYfuqQuUe9rvN2v_noxM7J51ng2BtZJ9wnXnKOBopz7SUnys-PJ1Ugig9T3UAJckgHvOQR2N9JqtYY7Q40aKrGIZAHs/s685/show.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"591\" data-original-width=\"685\" height=\"173\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6mQBB8l43_RfvbWygBnf7uTikJJnSTuKD68qTFW9FK4Qphb5_sYfuqQuUe9rvN2v_noxM7J51ng2BtZJ9wnXnKOBopz7SUnys-PJ1Ugig9T3UAJckgHvOQR2N9JqtYY7Q40aKrGIZAHs/w200-h173/show.png\" width=\"200\" /\u003e\u003c/a\u003e\nTo be able to use it in schools, the next step is to add activities to the application, such as creating classes, making lesson schedules, entering teachers and students and making lesson plans. That\u0026rsquo;s a glimpse of the information about Epelajar along with the menus and features found in the Epelajar application. I hope this is helpful. 😊.\u003c/li\u003e\n\u003c/ol\u003e\n", "language": "en" }, { "title" : "Learning Resources", "date_published" : "2020-09-10T15:38:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/perangkat-pembelajaran/", "url" : "https://www.aantriono.com/en/perangkat-pembelajaran/", "summary": " Learning Tools and Teacher Workload\nIn Minister of Education and Culture Regulation Number 15 of 2018 concerning Fulfilling the workload of Teachers, School Principals and School Supervisors dated 23 May 2018, in Article 3 paragraph 1 it is stated:\n(1) Implementation of workload for 37.5 (thirty seven point five) Effective working hours as intended in Article 2 paragraph (2) for Teachers include the main activities:\na. planning learning or mentoring;\nb. carrying out learning or mentoring;\nc. assessing learning or mentoring outcomes;\nd. guiding and training students; And\ne. carry out additional tasks attached to the implementation of activities principally according to the teacher's workload. ", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"learning-tools-and-teacher-workload\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-tools-and-teacher-workload\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Tools and Teacher Workload\"\u003e\u003c/a\u003eLearning Tools and Teacher Workload\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqMdm-oVlbbMI46-0R_2MkYVmAreWlcm6PlBCpDNH3X90puR41TtRcaoAYMv0LOHwNykbxPNaFceyc7nJFpJP3RIzEdZ6w4N0_FFbt5GQugD-u8fL2ee5Zr_9_y1HkaOrTJutzvdSrZPs/s485/pr.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Cover perangkat pembelajaran\" border=\"0\" data-original-height=\"485\" data-original-width=\"327\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqMdm-oVlbbMI46-0R_2MkYVmAreWlcm6PlBCpDNH3X90puR41TtRcaoAYMv0LOHwNykbxPNaFceyc7nJFpJP3RIzEdZ6w4N0_FFbt5GQugD-u8fL2ee5Zr_9_y1HkaOrTJutzvdSrZPs/s16000/pr.png\" title=\"Gambar Cover Perangkat Pembelajaran Guru\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003eIn Minister of Education and Culture Regulation Number 15 of 2018 concerning Fulfilling the workload of Teachers, School Principals and School Supervisors dated 23 May 2018, in Article 3 paragraph 1 it is stated:\u003c/p\u003e\n\u003cp\u003e(1) Implementation of workload for 37.5 (thirty seven point five) Effective working hours as intended in Article 2 paragraph (2) for Teachers include the main activities:\u003c/p\u003e\n\u003cp\u003ea. planning learning or mentoring;\u003c/p\u003e\n\u003cp\u003eb. carrying out learning or mentoring;\u003c/p\u003e\n\u003cp\u003ec. assessing learning or mentoring outcomes;\u003c/p\u003e\n\u003cp\u003ed. guiding and training students; And\u003c/p\u003e\n\u003cp\u003ee. carry out additional tasks attached to the implementation of activities principally according to the teacher's workload. \u003c/p\u003e\n\u003c!--more--\u003e\n\u003ch2 id=\"semester-program-and-tool-download\" class=\"headerLink\"\u003e\n \u003ca href=\"#semester-program-and-tool-download\" class=\"header-mark\" aria-label=\"Direct link to section: Semester Program and Tool Download\"\u003e\u003c/a\u003eSemester Program and Tool Download\u003c/h2\u003e\u003cp\u003eIn planning learning or mentoring activities, the activities carried out by teachers are analyzing the curriculum, analyzing the relationship between KI and KD, analyzing effective weeks, compiling annual programs, compiling semester programs, compiling syllabi, and compiling lesson plans. Below we present an example of the semester program that we organize at our school. Please click the download button below this.\u003c/p\u003e\u003c/p\u003e\n\u003cp\u003e\u003ca class=\"button\" href=\"https://drive.google.com/file/d/1ooX_Y2SlZU5EbltF9dtoPIfgUsQKhgBc/view?usp=sharing\"\u003eDownloads\u003c/a\u003e\n\u003c/p\u003e\nThis is an article about teacher activities in planning learning. I hope this is helpful.", "language": "en" }, { "title" : "Numbers", "date_published" : "2020-09-10T03:33:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bilangan/", "url" : "https://www.aantriono.com/en/bilangan/", "summary": " Order of Integers and Fractions\nBasic Competencies3.1 Explain and determine the order of positive and negative integers, as well as common fractions, mixed fractions, decimals, and percentages.\n3.2 Explain and perform arithmetic operations on whole numbers and fractions by using various operational properties.\n3.3 Explain and determine the representation of numbers in the form of positive and negative integers.\n4.1 Solve problems related to the order of integers and common fractions, mixed fractions, decimals, and percentages.\n4.2 Solve problems related to arithmetic operations on integers and fractions.\n4.3 Solve problems related to numbers in the form of positive and negative integers.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"order-of-integers-and-fractions\" class=\"headerLink\"\u003e\n \u003ca href=\"#order-of-integers-and-fractions\" class=\"header-mark\" aria-label=\"Direct link to section: Order of Integers and Fractions\"\u003e\u003c/a\u003eOrder of Integers and Fractions\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjioSzOlpwuAkVaWpUrx3UIl9yV1ZrNjrP_joNjhjuyMM9EWKHIciVCVUR7EuEMKxf_NRjuZC_5oUKEw81wgWkhhttajC4Hw7NdXl7erClhyy0ysz9gMLsbIj-k8jfl2sv9kUDpjjF2fng/s495/bb.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Gambar materi bilngan kelas VII SMP/MTs\" border=\"0\" data-original-height=\"367\" data-original-width=\"495\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjioSzOlpwuAkVaWpUrx3UIl9yV1ZrNjrP_joNjhjuyMM9EWKHIciVCVUR7EuEMKxf_NRjuZC_5oUKEw81wgWkhhttajC4Hw7NdXl7erClhyy0ysz9kMLsbIj-k8jfl2sv9kUDpjjF2fng/s16000/bb.png\" title=\"Gambar Bilangan\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h2\u003e\u003cp\u003e3.1 Explain and determine the order of positive and negative integers, as well as common fractions, mixed fractions, decimals, and percentages.\u003cbr\u003e\n3.2 Explain and perform arithmetic operations on whole numbers and fractions by using various operational properties.\u003cbr\u003e\n3.3 Explain and determine the representation of numbers in the form of positive and negative integers.\u003cbr\u003e\n4.1 Solve problems related to the order of integers and common fractions, mixed fractions, decimals, and percentages.\u003cbr\u003e\n4.2 Solve problems related to arithmetic operations on integers and fractions.\u003cbr\u003e\n4.3 Solve problems related to numbers in the form of positive and negative integers.\u003c/p\u003e\n\u003ch2 id=\"concept-of-integers-and-fractions\" class=\"headerLink\"\u003e\n \u003ca href=\"#concept-of-integers-and-fractions\" class=\"header-mark\" aria-label=\"Direct link to section: Concept of Integers and Fractions\"\u003e\u003c/a\u003eConcept of Integers and Fractions\u003c/h2\u003e\u003cp\u003eIn mathematics at SMP/MTs grade VII, number material is the first topic taught in the odd semester, consisting of whole numbers and fractions. Numbers themselves are a mathematical concept used for counting and measuring. Symbols used to represent a number are called numerals or number symbols. In mathematics, the concept of number has expanded over time to include zero, negative numbers, rational numbers, irrational numbers, and complex numbers. To better understand the concept of whole numbers and fractions, please read the following material presented in slide form.\u003c/p\u003e\n\u003cp\u003e\n\u003ciframe allowfullscreen=\"true\" frameborder=\"0\" height=\"300\" mozallowfullscreen=\"true\" src=\"https://docs.google.com/presentation/d/e/2PACX-1vRkFLvk6WTVz-ftGyjl5d5SascEcj3vcLWKOPKCJ94luFyK3LhjI5zQfo_NFO7YVg/embed?start=true\u0026loop=true\u0026delayms=3000\" webkitallowfullscreen=\"true\" width=\"450\"\u003e\u003c/iframe\u003e\n\u003c/p\u003e\n\u003cp\u003eThis is the number material in class VII SMP/MTs odd semester mathematics lessons. I hope this is helpful. 😊\u003c/p\u003e", "language": "en" }, { "title" : "How to Reset WordPress Password on Localhost", "date_published" : "2020-09-10T02:07:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/cara-mereset-password-wordpress-di/", "url" : "https://www.aantriono.com/en/cara-mereset-password-wordpress-di/", "summary": " Resetting WordPress Password Have you ever experienced an incident like the one in the picture above? when you want to log in to WordPress but forget the password 😇. Of course this is not desirable, so how do you reset the password? On this occasion, the author will share a tutorial for recovering WordPress passwords on localhost. So it\u0026rsquo;s not for WordPress which is already online huh 😊.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"resetting-wordpress-password\" class=\"headerLink\"\u003e\n \u003ca href=\"#resetting-wordpress-password\" class=\"header-mark\" aria-label=\"Direct link to section: Resetting WordPress Password\"\u003e\u003c/a\u003eResetting WordPress Password\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEig_0Xd4ifzg8oL8pzHpQ8CxGXstbyRXWpoTnsfsraxXl5kx0Vcsa8xTdTVBKTJY8zxsERpxqeTzbnb4DVYgxNCDhdkkOZFKRWjkZHuD9hUe11Vsrm6PzSWoxyVkSEaAIUiA587r7fwsuI/s522/xampp0.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Gambar lupa password di wordpress offline dan online\" border=\"0\" data-original-height=\"522\" data-original-width=\"414\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEig_0Xd4ifzg8oL8pzHpQ8CxGXstbyRXWpoTnsfsraxXl5kx0Vcsa8xTdTVBKTJY8zxsERpxqeTzbnb4DVYgxNCDhdkkOZFKRWjkZHuD9hUe11Vsrm6PzSWoxyVkSEaAIUiA587r7fwsuI/w254-h320/xampp0.png\" title=\"Gambar Lupa Password di Wordpress\" width=\"254\" /\u003e\u003c/a\u003e\nHave you ever experienced an incident like the one in the picture above? when you want to log in to WordPress but forget the password 😇. Of course this is not desirable, so how do you reset the password? On this occasion, the author will share a tutorial for recovering WordPress passwords on localhost. So it\u0026rsquo;s not for WordPress which is already online huh 😊.\u003c/p\u003e\n\u003ch2 id=\"steps-to-recover-password-on-localhost\" class=\"headerLink\"\u003e\n \u003ca href=\"#steps-to-recover-password-on-localhost\" class=\"header-mark\" aria-label=\"Direct link to section: Steps to Recover Password on Localhost\"\u003e\u003c/a\u003eSteps to Recover Password on Localhost\u003c/h2\u003e\u003cp\u003eThe steps are as follows:\u003c/p\u003e\n\u003cp\u003e1. Activate Localhost/XAMPP \u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYfevXDC32xS7aTuLWENu7UvRAU5-FDuiUbFqoHbl1rWDbXV-RLEPtz2xLVYCjuvwGtGt1lv7RhyphenhyphenMED6KKMl0j4miw_c-YBKuo_B6y_1SUUGCwLOF6Ozsa6K04gCLj8VltzEYiLnEuUeI/s670/xampp1.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"432\" data-original-width=\"670\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYfevXDC32xS7aTuLWENu7UvRAU5-FDuiUbFqoHbl1rWDbXV-RLEPtz2xLVYCjuvwGtGt1lv7RhyphenhyphenMED6KKMl0j4miw_c-YBKuo_B6y_1SUUGCwLOF6Ozsa6K04gCLj8VltzEYiLnEuUeI/s320/xampp1.png\" width=\"320\" /\u003e\u003c/a\u003e\n\u003cp\u003e2. Access your WordPress database by typing the url localhost/phpmyadmin\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIeDGuKnNo84pZQ9kLUifN6N1Ic8Rdn4yXN8R7xNsyF4h0dVqBGv4OHUKmcz2VZbAA4KNsDYNTjNADmhSGttLC50LjYNTW5YY7iIqP-X3XabFWaDPhyUUt3Rcyj3qCq-bnGKlMIY8pKxg/s1165/xampp2.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"559\" data-original-width=\"1165\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIeDGuKnNo84pZQ9kLUifN6N1Ic8Rdn4yXN8R7xNsyF4h0dVqBGv4OHUKmcz2VZbAA4KNsDYNTjNADmhSGttLC50LjYNTW5YY7iIqP-X3XabFWaDPhyUUt3Rcyj3qCq-bnGKlMIY8pKxg/s320/xampp2.png\" width=\"320\" /\u003e\u003c/a\u003e\n\u003cp\u003e3. Look for your database name in the left menu of the phpmyadmin page\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJPR68tS7pT-DfwCXbsf6ej4_cmMDZvjhAuI2owVcLzQzeaia285wPv6-Nd4nXpdw4cHWQHHZWg9gQVfrLoEyBEqJ66fuhqTrB2M8Rc2lpB_OTjJa8DDgo823L8uHlvgVqSWQkKIXc_zo/s1178/xampp3.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"692\" data-original-width=\"1178\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJPR68tS7pT-DfwCXbsf6ej4_cmMDZvjhAuI2owVcLzQzeaia285wPv6-Nd4nXpdw4cHWQHHZWg9gQVfrLoEyBEqJ66fuhqTrB2M8Rc2lpB_OTjJa8DDgo823L8uHlvgVqSWQkKIXc_zo/s320/xampp3.png\" width=\"320\" /\u003e\u003c/a\u003e\n\u003cp\u003e4. Click _users and look for your login, if you have found it, please click edit as shown in the following image. The user_pass box will be edited\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFdk6Pvr2UzVMSYPho0cW45JaqN6oahqmVRyqTsFSsHM3xH6MpkRxLuhxuiaYH1Ss1aYZsptSFdZ815O6fVAt2maTTUHanaFSsAWRvcdokrW1Sule0157RofIRenVx_emGpqcRilRGFb0/s927/xampp4.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"423\" data-original-width=\"927\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFdk6Pvr2UzVMSYPho0cW45JaqN6oahqmVRyqTsFSsHM3xH6MpkRxLuhxuiaYH1Ss1aYZsptSFdZ815O6fVAt2maTTUHanaFSsAWRvcdokrW1Sule0157RofIRenVx_emGpqcRilRGFb0/s320/xampp4.png\" width=\"320\" /\u003e\u003c/a\u003e\n\u003cp\u003e5. In the user_pass drop down box, look for the word MD5 and change the password in the input text box next to it according to your wishes.\u003c/p\u003e\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhj1SxosmBY9V48055GaYi20ZoQMaKDW0qPOXcvs7zFds1rvjCUvnQ4Ee0_ay22T9hPOLpIJ6qSd6_URIkcFbgMn-XW6PnE7uBdJHPgQeHCSuyBp7mKm2KAlEnw6OxlD-8CEia5XWWh2nI/s937/xampp5.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"354\" data-original-width=\"937\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhj1SxosmBY9V48055GaYi20ZoQMaKDW0qPOXcvs7zFds1rvjCUvnQ4Ee0_ay22T9hPOLpIJ6qSd6_URIkcFbgMn-XW6PnE7uBdJHPgQeHCSuyBp7mKm2KAlEnw6OxlD-8CEia5XWWh2nI/s320/xampp5.png\" width=\"320\" /\u003e\u003c/a\u003e\n\u003cp\u003e6. If it has been changed, please scroll down and look for the word Go then click.\n\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgwPyhL0q6S6TXLnPVRTGDIjRRd7VZ7jmIbrRZZWlFFw_W3kZEAuzxCNrhmT6N7OHdhRxH3CjGeUc43A9froxSUD7mshFhx9EbeaD5HSQMBwinWO4ggyCVnpsixAS1kuicTbaw8seNVD8/s744/xampp6.png\" style=\"font-family: verdana; margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"122\" data-original-width=\"744\" height=\"81\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgwPyhL0q6S6TXLnPVRTGDIjRRd7VZ7jmIbrRZZWlFFw_W3kZEAuzxCNrhmT6N7OHdhRxH3CjGeUc43A9froxSUD7mshFhx9EbeaD5HSQMBwinWO4ggyCVnpsixAS1kuicTbaw8seNVD8/w500-h81/xampp6.png\" width=\"500\" /\u003e\u003c/a\u003e\n\u003c/p\u003e\nAfter these steps have been carried out, please access your WordPress I page again, and type in the user and password with I user and I password.\n\u003cp\u003eThat's the article about how to restore WordPress passwords offline or on localhost. I hope this is helpful. and you can start working again with WordPress on your local server. Good luck, always success 😊. Thank you.\u003c/p\u003e", "language": "en" }, { "title" : "Mahabbah, Part 2", "date_published" : "2020-09-09T09:20:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mahabbah2/", "url" : "https://www.aantriono.com/en/mahabbah2/", "summary": " Intuition and the Criteria of Loving AllahImage of love Love is about human intuition. The feeling of love is not determined by social level or race — it is universal. There is no absolute criterion for love. What is clear is that love for Allah, or the coming of Allah\u0026rsquo;s love to human beings, is completely dependent on masyiatullah — Allah\u0026rsquo;s will.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"intuition-and-the-criteria-of-loving-allah\" class=\"headerLink\"\u003e\n \u003ca href=\"#intuition-and-the-criteria-of-loving-allah\" class=\"header-mark\" aria-label=\"Direct link to section: Intuition and the Criteria of Loving Allah\"\u003e\u003c/a\u003eIntuition and the Criteria of Loving Allah\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCEfjgt6OJ22uOIJ3w8EG6X3N9VIJiEpSYjgKB-r9icmDzLslnMM1vgyeshuc7Zrg6yAePxO6iW6oTKg76z7ZVMS1zqWu4hj81QG1zyW-nTPO5Xk0xo9_gMT5o_etW2qhVQxkqHZFnq24/w320-h213/ci.jpg\" title=\"the lovers of Allah\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCEfjgt6OJ22uOIJ3w8EG6X3N9VIJiEpSYjgKB-r9icmDzLslnMM1vgyeshuc7Zrg6yAePxO6iW6oTKg76z7ZVMS1zqWu4hj81QG1zyW-nTPO5Xk0xo9_gMT5o_etW2qhVQxkqHZFnq24/w320-h213/ci.jpg\" data-sub-html=\"\u003ch2\u003eImage of love\u003c/h2\u003e\u003cp\u003ethe lovers of Allah\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCEfjgt6OJ22uOIJ3w8EG6X3N9VIJiEpSYjgKB-r9icmDzLslnMM1vgyeshuc7Zrg6yAePxO6iW6oTKg76z7ZVMS1zqWu4hj81QG1zyW-nTPO5Xk0xo9_gMT5o_etW2qhVQxkqHZFnq24/w320-h213/ci.jpg' alt=\"the lovers of Allah\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eImage of love\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eLove is about human intuition. The feeling of love is not determined by social level or race — it is universal. There is no absolute criterion for love. What is clear is that love for Allah, or the coming of Allah\u0026rsquo;s love to human beings, is completely dependent on \u003cem\u003emasyiatullah\u003c/em\u003e — Allah\u0026rsquo;s will.\u003c/p\u003e\n\u003cp\u003eImam Ibn \u0026lsquo;Athaillah, when asked about Mahabbah, answered: \u0026ldquo;It is a branch that grows in the heart and then develops and bears fruit according to the level of intelligence.\u0026rdquo; This relates to how one expresses love. Someone who is intelligent will be better able to express their love than someone with below-average intelligence. Or in experiencing love, someone with higher intelligence will certainly have more mature thoughts than someone below them. This can also happen due to differences in language ability. A poet is certainly more poetic and touching in expressing love than someone who rarely works with language.\u003c/p\u003e\n\u003ch2 id=\"levels-of-love-in-language-and-philosophy\" class=\"headerLink\"\u003e\n \u003ca href=\"#levels-of-love-in-language-and-philosophy\" class=\"header-mark\" aria-label=\"Direct link to section: Levels of Love in Language and Philosophy\"\u003e\u003c/a\u003eLevels of Love in Language and Philosophy\u003c/h2\u003e\u003cp\u003eWhen the natural inclination (\u003cem\u003emahabbah\u003c/em\u003e) is quite strong, it is called \u003cstrong\u003eshobabah\u003c/strong\u003e — an outpouring of love, because of the total pouring out of affection. When the inclination grows stronger, it is called \u003cstrong\u003eghoram\u003c/strong\u003e — a steady love born of certainty. When it becomes even stronger, it is called \u003cstrong\u003e\u0026lsquo;isq\u003c/strong\u003e — love that goes beyond limits. When stronger still, it is called \u003cstrong\u003esyagaf\u003c/strong\u003e, because it reaches the \u003cem\u003esyaghaful qalb\u003c/em\u003e — the innermost layer of the heart, driven from within.\u003c/p\u003e\n\u003ch2 id=\"three-categories-of-lovers-of-allah\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-categories-of-lovers-of-allah\" class=\"header-mark\" aria-label=\"Direct link to section: Three Categories of Lovers of Allah\"\u003e\u003c/a\u003eThree Categories of Lovers of Allah\u003c/h2\u003e\u003cp\u003eKnow that those who love Allah (\u003cem\u003emuhibbin\u003c/em\u003e) consist of three groups:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eThe common people (\u003cem\u003eawam\u003c/em\u003e)\u003c/strong\u003e — those who love Allah Ta\u0026rsquo;ala in order to attain His goodness.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe special ones (\u003cem\u003ekhawwash\u003c/em\u003e)\u003c/strong\u003e — their love is sincere, pure, and free from any mixtures or ulterior motives.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe most special ones (\u003cem\u003ekhawwasul-khawwash\u003c/em\u003e)\u003c/strong\u003e — their love is described as a deep longing, through which the one who longs is themselves erased when the Beloved they long for becomes manifest.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eExcerpted from the book \u003cem\u003eTanwirul Qulub\u003c/em\u003e.\u003c/p\u003e", "language": "en" }, { "title" : "Relations and Functions", "date_published" : "2020-09-08T14:59:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/relasi-dan-fungsi/", "url" : "https://www.aantriono.com/en/relasi-dan-fungsi/", "summary": " Introduction\nBasic Competencies Describe and represent relations and functions using words, tables, graphs, diagrams, and equations. Solve problems related to relations and functions using different representations. Learning Experience Explain everyday situations related to relations and functions. Define relations and functions. Understand the difference between a relation and a non-relation. Identify functions and non-functions. Understand different ways of representing relations and functions. Draw function graphs on the Cartesian plane. ", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6A_lX2Yq45ahx7x46qBzk2fiwLzuME5qdzaDhZaIe8hvqokMy7y76Rr0Jjm3fJhJum3tZRLclaEYGxqhlMkJZTg6iVI0fW9f0NLz35vqSyiIZYjxQkscOoEHKOMoC1fsqW6hOL6tBeWw/w320-h240/relas.jpg' alt=\"Relations and functions\" \u003e\u003c/p\u003e\n\u003ch2 id=\"basic-competencies\" class=\"headerLink\"\u003e\n \u003ca href=\"#basic-competencies\" class=\"header-mark\" aria-label=\"Direct link to section: Basic Competencies\"\u003e\u003c/a\u003eBasic Competencies\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eDescribe and represent relations and functions using words, tables, graphs, diagrams, and equations.\u003c/li\u003e\n\u003cli\u003eSolve problems related to relations and functions using different representations.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"learning-experience\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-experience\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Experience\"\u003e\u003c/a\u003eLearning Experience\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eExplain everyday situations related to relations and functions.\u003c/li\u003e\n\u003cli\u003eDefine relations and functions.\u003c/li\u003e\n\u003cli\u003eUnderstand the difference between a relation and a non-relation.\u003c/li\u003e\n\u003cli\u003eIdentify functions and non-functions.\u003c/li\u003e\n\u003cli\u003eUnderstand different ways of representing relations and functions.\u003c/li\u003e\n\u003cli\u003eDraw function graphs on the Cartesian plane.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eIn daily life, we often hear the word \u003cem\u003erelation\u003c/em\u003e or \u003cem\u003econnection\u003c/em\u003e, such as family relationships or business relations. In mathematics, a relation connects two groups of objects according to a certain rule.\u003c/p\u003e\n\u003cp\u003eTo study the material more systematically, you can use the following presentation slides.\u003c/p\u003e\n\u003ciframe src=\"https://docs.google.com/presentation/d/e/2PACX-1vT4FxPxEjw4AYqBX34Ul6JNsW0njS-lUjxWjkQhJd-8nRo2DwYGKVP3oHrXzXVM4g/embed?start=true\u0026loop=true\u0026delayms=3000\" frameborder=\"0\" width=\"650\" height=\"300\" allowfullscreen=\"true\" mozallowfullscreen=\"true\" webkitallowfullscreen=\"true\"\u003e\u003c/iframe\u003e\n\u003cp\u003eFor additional explanation, watch the following video.\u003c/p\u003e\n\u003ciframe allowfullscreen=\"\" frameborder=\"0\" height=\"300\" src=\"https://www.youtube.com/embed/OTq4-SBshAA\" width=\"650\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThat concludes this material on relations and functions. I hope it is useful.\u003c/p\u003e", "language": "en" }, { "title" : "Assessment of Quadratic Equation Material", "date_published" : "2020-09-08T06:01:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/penilaian-materi-persamaan-kuadrat/", "url" : "https://www.aantriono.com/en/penilaian-materi-persamaan-kuadrat/", "summary": " Introduction\nAfter studying quadratic equations, it is time to do an assessment to measure competency achievement. This quiz was originally prepared for students of SMPN Purworejo, Padang Ratu District, Central Lampung.\nOther readers are also welcome to try the quiz or share it with their students.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg11MUYCcrWjIMxg3yYYSFLGv9NBn05eHJEyqBMY9k9fSCAj-mYXF8h2f6o5YNmS5L-8m6NVbaKPSMg4y3mAGNR5bAYPKxt0wsdFt2UiM-vBUjoAYPuIaJw3V-0gWIeTIFD_0UmVHmpXHI/w640-h490/SOAL+PENILAIAN+HARIAN.png' alt=\"Quadratic equation daily assessment\" \u003e\u003c/p\u003e\n\u003cp\u003eAfter studying quadratic equations, it is time to do an assessment to measure competency achievement. This quiz was originally prepared for students of SMPN Purworejo, Padang Ratu District, Central Lampung.\u003c/p\u003e\n\u003cp\u003eOther readers are also welcome to try the quiz or share it with their students.\u003c/p\u003e\n\u003ch2 id=\"instructions\" class=\"headerLink\"\u003e\n \u003ca href=\"#instructions\" class=\"header-mark\" aria-label=\"Direct link to section: Instructions\"\u003e\u003c/a\u003eInstructions\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eThe quiz contains 10 questions with \u003ccode\u003eMultiple Choice\u003c/code\u003e, \u003ccode\u003eCheckbox\u003c/code\u003e, and \u003ccode\u003eTrue/False\u003c/code\u003e formats.\u003c/li\u003e\n\u003cli\u003eThe time limit is 30 minutes.\u003c/li\u003e\n\u003cli\u003eThe minimum passing criterion is 60.\u003c/li\u003e\n\u003cli\u003eWrite your first name, last name, and email in the provided fields before starting.\u003c/li\u003e\n\u003cli\u003eIf you only have one name, fill the last-name field with your class name. Example: first name \u003ccode\u003eAndi\u003c/code\u003e, last name \u003ccode\u003e9B\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eIf you have both first and last names, add the class name after the last name. Example: \u003ccode\u003eAslam Permana 9B\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eIf you do not have an email address, use a simple format such as \u003ccode\u003eandi@gmail.com\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eClick \u003ccode\u003eStart\u003c/code\u003e to begin.\u003c/li\u003e\n\u003cli\u003eAnswer each question by choosing the correct option.\u003c/li\u003e\n\u003cli\u003eFor \u003ccode\u003eCheckbox\u003c/code\u003e questions, choose 2 or 3 answers as required.\u003c/li\u003e\n\u003cli\u003eAfter answering all questions, click \u003ccode\u003eSubmit my answer\u003c/code\u003e.\u003c/li\u003e\n\u003cli\u003eAfter completion, a certificate and score report will appear.\u003c/li\u003e\n\u003cli\u003eThe certificate can be downloaded, and the score can be saved with a screenshot as proof of completion.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"quiz\" class=\"headerLink\"\u003e\n \u003ca href=\"#quiz\" class=\"header-mark\" aria-label=\"Direct link to section: Quiz\"\u003e\u003c/a\u003eQuiz\u003c/h2\u003e\u003ciframe frameborder=\"0\" height=\"300\" id=\"proprofs\" marginheight=\"0\" marginwidth=\"0\" name=\"proprofs\" src=\"https://www.proprofs.com/quiz-school/story.php?title=penilaian-materi-persamaan-kuadrat\u0026id=2875795\u0026ew=430\" width=\"450\"\u003e\u003c/iframe\u003e\n\u003cp\u003e\u003ca href=\"https://www.proprofs.com/quiz-school/story.php?title=penilaian-materi-persamaan-kuadrat\" target=\"_blank\" rel=\"noopener noreferrer\"\u003eOpen the quiz on ProProfs\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#closing\" class=\"header-mark\" aria-label=\"Direct link to section: Closing\"\u003e\u003c/a\u003eClosing\u003c/h2\u003e\u003cp\u003eI hope this simple assessment is useful for teachers, students, and education in general.\u003c/p\u003e\n", "language": "en" }, { "title" : "Mahabbah", "date_published" : "2020-09-07T15:40:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mahabbah/", "url" : "https://www.aantriono.com/en/mahabbah/", "summary": " The Concept of Mahabbah and Divine LoveLove Speaking about mahabbah, or love, is always compelling. Some say that the rope of Allah stretched across the universe is love itself. Those who speak of hubbul ilahi, the love of God — including thinkers such as Ibn Sina — hold that everything in creation is touched by love. In their view, the beings created by Allah exist in pairs. Fire and water are not merely opposites; they can also be understood as companions within the order of creation. When a great fire burns, it seems to call for water, and when water comes, smoke rises into the sky as if the meeting of the two has produced its own powerful symphony.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"the-concept-of-mahabbah-and-divine-love\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-concept-of-mahabbah-and-divine-love\" class=\"header-mark\" aria-label=\"Direct link to section: The Concept of Mahabbah and Divine Love\"\u003e\u003c/a\u003eThe Concept of Mahabbah and Divine Love\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqv029mvyk3A6BtbopH7ffWy6wkXk-nJw8a-WV0s6b6oiiYyyaG3Y_bt8Tfkunr-t7nkvKQTlWb34dYWrHqMgTXM6uKGQ_EM-5yE1lRb0tBFiX55SWsEKP5PR8zbPL7w2kY4yxDJAMeOQ/w320-h213/ci.jpg\" title=\"When love meets\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqv029mvyk3A6BtbopH7ffWy6wkXk-nJw8a-WV0s6b6oiiYyyaG3Y_bt8Tfkunr-t7nkvKQTlWb34dYWrHqMgTXM6uKGQ_EM-5yE1lRb0tBFiX55SWsEKP5PR8zbPL7w2kY4yxDJAMeOQ/w320-h213/ci.jpg\" data-sub-html=\"\u003ch2\u003eLove\u003c/h2\u003e\u003cp\u003eWhen love meets\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqv029mvyk3A6BtbopH7ffWy6wkXk-nJw8a-WV0s6b6oiiYyyaG3Y_bt8Tfkunr-t7nkvKQTlWb34dYWrHqMgTXM6uKGQ_EM-5yE1lRb0tBFiX55SWsEKP5PR8zbPL7w2kY4yxDJAMeOQ/w320-h213/ci.jpg' alt=\"When love meets\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eLove\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eSpeaking about \u003cem\u003emahabbah\u003c/em\u003e, or love, is always compelling. Some say that the rope of Allah stretched across the universe is love itself. Those who speak of \u003cem\u003ehubbul ilahi\u003c/em\u003e, the love of God — including thinkers such as Ibn Sina — hold that everything in creation is touched by love. In their view, the beings created by Allah exist in pairs. Fire and water are not merely opposites; they can also be understood as companions within the order of creation. When a great fire burns, it seems to call for water, and when water comes, smoke rises into the sky as if the meeting of the two has produced its own powerful symphony.\u003c/p\u003e\n\u003ch2 id=\"the-universality-of-love\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-universality-of-love\" class=\"header-mark\" aria-label=\"Direct link to section: The Universality of Love\"\u003e\u003c/a\u003eThe Universality of Love\u003c/h2\u003e\u003cp\u003eShaykh Amin al-Kurdi defines love as a natural inclination toward something that brings deep delight to the one who loves. Love gives rise to joy, while separation from love often becomes the source of sorrow.\u003c/p\u003e\n\u003cp\u003eBecause love carries beauty and sweetness, it has inspired many forms of art, expression, and longing throughout human life. The sense of beauty placed within the human being awakens attraction, and from that attraction love begins to grow.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTo be continued\u0026hellip;\u003c/strong\u003e\u003c/p\u003e", "language": "en" }, { "title" : "Online Questions", "date_published" : "2020-09-06T15:55:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/soal-online/", "url" : "https://www.aantriono.com/en/soal-online/", "summary": " Creating and Integrating Online Assessments\nAs an educator, there are many ways to keep teaching and helping students continue studying at home during the Covid-19 pandemic. The learning media used to deliver lesson material and assessments also vary. Some use WhatsApp, Telegram, Google Form, Google Classroom, LMS Moodle, Edmodo, blogs, and more.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"creating-and-integrating-online-assessments\" class=\"headerLink\"\u003e\n \u003ca href=\"#creating-and-integrating-online-assessments\" class=\"header-mark\" aria-label=\"Direct link to section: Creating and Integrating Online Assessments\"\u003e\u003c/a\u003eCreating and Integrating Online Assessments\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfmj7wyELQJsrlYIRoukQEmAKsyVargVp9kR_NtD-LiVRczLZsQK6wlR6ucLPUAgOqwTg8-GLgIhwvFbbUFKF4wutRM7OjFYhkJAqq2L8-Qe0z761_R_rAlkpcSL5eEN_hCjBGSALFsCs/s16000/images.jpg' alt=\"Online math questions for class 9 SMP\" \u003e\u003c/p\u003e\n\u003cp\u003eAs an educator, there are many ways to keep teaching and helping students continue studying at home during the Covid-19 pandemic. The learning media used to deliver lesson material and assessments also vary. Some use WhatsApp, Telegram, Google Form, Google Classroom, LMS Moodle, Edmodo, blogs, and more.\u003c/p\u003e\n\u003ch2 id=\"applications-for-online-question-creation\" class=\"headerLink\"\u003e\n \u003ca href=\"#applications-for-online-question-creation\" class=\"header-mark\" aria-label=\"Direct link to section: Applications for Online Question Creation\"\u003e\u003c/a\u003eApplications for Online Question Creation\u003c/h2\u003e\u003cp\u003eAll applications used to convey study material or to make assignments/assessments online certainly have their respective advantages and disadvantages. Therefore, of course it must be adjusted to the readiness and needs of the educators and students. But there is no harm as an educator in trying various applications, especially for creating online assessment questions. Everything is worth trying not only to improve the professional competence of educators but also to meet the needs of the students themselves. There are several applications for creating questions offline and online, such as Wondershare Quizmaker, Ispring Quizmaker, ExamView and so on. On this occasion the author tried to create questions online using online question maker website and embed it into the blog. The website is equipped with a certificate if you have passed or the score is above the KKM/Minimum Completion Criteria. What are the results like? please see below.\u003c/p\u003e\n\u003cblockquote\u003e\n \u003cp\u003eThis interactive quiz example is no longer available on ProProfs (removed by the service provider).\u003c/p\u003e\n\u003c/blockquote\u003e\u003cp\u003eThis is the appearance of the questions created online using the online question maker website. I hope this is helpful.\u003c/p\u003e", "language": "en" }, { "title" : "The First Teaching", "date_published" : "2020-09-05T14:51:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/ajaran-pertama/", "url" : "https://www.aantriono.com/en/ajaran-pertama/", "summary": " First Teaching of Futuhul GhaibTasawuf Biography of Sheikh Abdul Qadir al-JailaniSayyid Abu Muhammad Abdul Qadir was born in Naif, in the region of Jilan, Persia, in Ramadan 470 AH, or around 1077 CE. His father, Abu Shaleh, was known for his devotion and traced his lineage to Imam Hasan, the eldest son of Ali ibn Abi Talib and Fatimah al-Zahra. His mother was the daughter of Abdullah Shaumayya, a devout woman descended from Imam Husain. In that sense, Sheikh Abdul Qadir al-Jailani is remembered as carrying the lineage of both Imam Hasan and Imam Husain.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"first-teaching-of-futuhul-ghaib\" class=\"headerLink\"\u003e\n \u003ca href=\"#first-teaching-of-futuhul-ghaib\" class=\"header-mark\" aria-label=\"Direct link to section: First Teaching of Futuhul Ghaib\"\u003e\u003c/a\u003eFirst Teaching of Futuhul Ghaib\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQp_eWZCVjZRn-ms_NZTdFsQksOu-YaCOzBIUGOvyEyeVulbrF-6X96b6JSQRQLDaAvkUDsUbi2hmMNwkQDpmEwB2ZLT4VBpLtMD7zIQRlpOPaalwAwYIt2f4OOh9wSh3YG96PYaRUc8I/s16000/Tasawuf_Ilmu_yang_Tak_Cukup.jpg\" title=\"Tasawuf in Islam\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQp_eWZCVjZRn-ms_NZTdFsQksOu-YaCOzBIUGOvyEyeVulbrF-6X96b6JSQRQLDaAvkUDsUbi2hmMNwkQDpmEwB2ZLT4VBpLtMD7zIQRlpOPaalwAwYIt2f4OOh9wSh3YG96PYaRUc8I/s16000/Tasawuf_Ilmu_yang_Tak_Cukup.jpg\" data-sub-html=\"\u003ch2\u003eTasawuf\u003c/h2\u003e\u003cp\u003eTasawuf in Islam\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQp_eWZCVjZRn-ms_NZTdFsQksOu-YaCOzBIUGOvyEyeVulbrF-6X96b6JSQRQLDaAvkUDsUbi2hmMNwkQDpmEwB2ZLT4VBpLtMD7zIQRlpOPaalwAwYIt2f4OOh9wSh3YG96PYaRUc8I/s16000/Tasawuf_Ilmu_yang_Tak_Cukup.jpg' alt=\"Tasawuf in Islam\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eTasawuf\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003ch2 id=\"biography-of-sheikh-abdul-qadir-al-jailani\" class=\"headerLink\"\u003e\n \u003ca href=\"#biography-of-sheikh-abdul-qadir-al-jailani\" class=\"header-mark\" aria-label=\"Direct link to section: Biography of Sheikh Abdul Qadir al-Jailani\"\u003e\u003c/a\u003eBiography of Sheikh Abdul Qadir al-Jailani\u003c/h2\u003e\u003cp\u003eSayyid Abu Muhammad Abdul Qadir was born in Naif, in the region of Jilan, Persia, in Ramadan 470 AH, or around 1077 CE. His father, Abu Shaleh, was known for his devotion and traced his lineage to Imam Hasan, the eldest son of Ali ibn Abi Talib and Fatimah al-Zahra. His mother was the daughter of Abdullah Shaumayya, a devout woman descended from Imam Husain. In that sense, Sheikh Abdul Qadir al-Jailani is remembered as carrying the lineage of both Imam Hasan and Imam Husain.\u003c/p\u003e\n\u003cp\u003eAs a scholar, Sheikh Abdul Qadir al-Jailani was known not only for his knowledge but also for the way he lived it out through guidance, legal opinions, and spiritual counsel. Three collections of his teachings became especially well known: \u003cem\u003eFutuhul Ghaib\u003c/em\u003e, \u003cem\u003eFathul Rabbani\u003c/em\u003e, and \u003cem\u003eQasidatul Ghautsiyyah\u003c/em\u003e. The book \u003cem\u003eFutuhul Ghaib\u003c/em\u003e contains eighty pieces of advice. The passage below presents the first of those teachings:\u003c/p\u003e\n\u003ch2 id=\"three-essential-devotions\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-essential-devotions\" class=\"header-mark\" aria-label=\"Direct link to section: Three Essential Devotions\"\u003e\u003c/a\u003eThree Essential Devotions\u003c/h2\u003e\u003cp\u003e\u003cem\u003eThere are three things that every believer must guard in every situation:\u003c/em\u003e\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003ecarrying out all of God\u0026rsquo;s commands;\u003c/li\u003e\n\u003cli\u003estaying away from what is forbidden;\u003c/li\u003e\n\u003cli\u003eaccepting God\u0026rsquo;s decrees with contentment.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThese three qualities should never be absent in a believer. A believer should reflect on them, examine himself through them, and let every part of his life be shaped by them.\u003c/p\u003e\n\u003cp\u003eThat is the opening teaching presented in \u003cem\u003eFutuhul Ghaib\u003c/em\u003e. May it be beneficial.\u003c/p\u003e", "language": "en" }, { "title" : "Cartesian coordinates", "date_published" : "2020-09-05T08:03:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/koordinat-cartesius/", "url" : "https://www.aantriono.com/en/koordinat-cartesius/", "summary": " Origin of the Cartesian Coordinate System The term Cartesius (read: Cartesius) is a latinization of Descartes. This term is used to commemorate the French mathematician and philosopher, Descartes, who played a major role in combining algebra and geometry. He introduced the new idea of ​​describing the position of a point or object on a surface using two axes that are perpendicular to each other. Cartesian coordinates are used to determine object points on a plane using two numbers which are usually called the x coordinate and y coordinate of the points. To define coordinates, two lines are needed perpendicular to each other, namely the X-axis and the Y-axis, and the unit length of which is made by markings on these two axes.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"origin-of-the-cartesian-coordinate-system\" class=\"headerLink\"\u003e\n \u003ca href=\"#origin-of-the-cartesian-coordinate-system\" class=\"header-mark\" aria-label=\"Direct link to section: Origin of the Cartesian Coordinate System\"\u003e\u003c/a\u003eOrigin of the Cartesian Coordinate System\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnYfncLkRpBy5gYznNVIrenzzeauJmy_47cNMud0jhWesP3iKlZX8oTt5xABX9w_TPBrR_snuUMPEhHO_6aGB70EGQHPHob5sDNvJ9zZxH7bR3HJKAG0BAdJT3rdcv0eO3pqP_7Y7Ih-4/s487/gkc.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"Cara menggambar Koordinat Kartesius\" border=\"0\" data-original-height=\"482\" data-original-width=\"487\" height=\"317\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnYfncLkRpBy5gYznNVIrenzzeauJmy_47cNMud0jhWesP3iKlZX8oTt5xABX9w_TPBrR_snuUMPEhHO_6aGB70EGQHPHob5sDNvJ9zZxH7bR3HJKAG0BAdJT3rdcv0eO3pqP_7Y7Ih-4/w320-h317/gkc.png\" title=\"Gambar koordinat kartesius\" width=\"320\" /\u003e\u003c/a\u003e\nThe term Cartesius (read: Cartesius) is a latinization of Descartes. This term is used to commemorate the French mathematician and philosopher, Descartes, who played a major role in combining algebra and geometry. He introduced the new idea of ​​describing the position of a point or object on a surface using two axes that are perpendicular to each other. Cartesian coordinates are used to determine object points on a plane using two numbers which are usually called the x coordinate and y coordinate of the points. To define coordinates, two lines are needed perpendicular to each other, namely the X-axis and the Y-axis, and the unit length of which is made by markings on these two axes.\u003c/p\u003e\n\u003ch2 id=\"quadrant-division-in-cartesian-coordinates\" class=\"headerLink\"\u003e\n \u003ca href=\"#quadrant-division-in-cartesian-coordinates\" class=\"header-mark\" aria-label=\"Direct link to section: Quadrant Division in Cartesian Coordinates\"\u003e\u003c/a\u003eQuadrant Division in Cartesian Coordinates\u003c/h2\u003e\u003cp\u003eLook at the following image:\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyyxcJPhLaQsbn6rejsghXiC1howdDOKTUol0pyJpWdPOjtzDliTPzRlFoHa-N-ycMKQbVi7-T3DtT1OzRQF5fDh7prbcnKoymvYSIlJLKfKyNwvb6Tp-A8goyuYfUVELpazAOGOOXUwA/s258/picture1+%25281%2529.png\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"257\" data-original-width=\"258\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyyxcJPhLaQsbn6rejsghXiC1howdDOKTUol0pyJpWdPOjtzDliTPzRlFoHa-N-ycMKQbVi7-T3DtT1OzRQF5fDh7prbcnKoymvYSIlJLKfKyNwvb6Tp-A8goyuYfUVELpazAOGOOXUwA/s0/picture1+%25281%2529.png\" /\u003e\u003c/a\u003e\nAccording to the applicable convention, the four quadrant areas are ordered starting from the top right, circular counterclockwise. In quadrant I, both x and y coordinates will be positive. In quadrant II, the x coordinate will be negative and the y coordinate will be positive. In quadrant III, both coordinates will be negative and in quadrant IV, the x coordinate will be positive and y will be negative. For more clarity, please see the following slide:\u003c/p\u003e\n\u003ciframe allowfullscreen=\"true\" frameborder=\"0\" height=\"500\" mozallowfullscreen=\"true\" src=\"https://docs.google.com/presentation/d/e/2PACX-1vQJB0CQs8wQk2sjCFMb9sUj_VLvgmotHjyCsEB7QMVr5Og71ZfbRwT09W0CW5lCmQ/embed?start=true\u0026loop=true\u0026delayms=3000\" webkitallowfullscreen=\"true\" width=\"600\"\u003e\u003c/iframe\u003e\nThat's the material about Cartesian coordinates. I hope this is helpful. 😊", "language": "en" }, { "title" : "Solving Quadratic Equations by Factoring and Completing Perfect Squares", "date_published" : "2020-09-03T14:16:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/penyelesaian-persamaan-kuadrat-dengan/", "url" : "https://www.aantriono.com/en/penyelesaian-persamaan-kuadrat-dengan/", "summary": " Introduction\nQuadratic equations can be solved in three main ways:\nFactoring, which changes the sum of algebraic terms into a product form. Completing the square, which changes the general form into a perfect-square form. The quadratic formula or abc formula, which gives the roots directly. This article focuses on solving quadratic equations by factoring and completing the square.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnn0ZmwTJUsA4zUc3Q-XzSztv-PHorZkEY4T9yzqVTLnI6Jc6C4zZjMyXjCglO_JHHlooDl9eIMVBHPO6_jjuflgWtMPMDa61nQnxUuoyrXvL1KKNCeFxzL04D-h-Ml-sFbpHRSxY2EMc/w640-h386/persamaan-kuadrat-630x380.png' alt=\"Quadratic equation\" \u003e\u003c/p\u003e\n\u003cp\u003eQuadratic equations can be solved in three main ways:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cstrong\u003eFactoring\u003c/strong\u003e, which changes the sum of algebraic terms into a product form.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eCompleting the square\u003c/strong\u003e, which changes the general form into a perfect-square form.\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eThe quadratic formula or abc formula\u003c/strong\u003e, which gives the roots directly.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eThis article focuses on solving quadratic equations by \u003cstrong\u003efactoring\u003c/strong\u003e and \u003cstrong\u003ecompleting the square\u003c/strong\u003e.\u003c/p\u003e\n\u003ch2 id=\"learning-video\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-video\" class=\"header-mark\" aria-label=\"Direct link to section: Learning Video\"\u003e\u003c/a\u003eLearning Video\u003c/h2\u003e\u003cp\u003eTo help you understand the material, watch the following video.\u003c/p\u003e\n\u003ciframe allowfullscreen=\"\" frameborder=\"0\" height=\"300\" src=\"https://www.youtube.com/embed/Zl0d8Reh5jY\" width=\"550\"\u003e\u003c/iframe\u003e\n\u003cp\u003eThat concludes this short discussion on solving quadratic equations by factoring and completing perfect squares.\u003c/p\u003e\n", "language": "en" }, { "title" : "The Ashura Fast", "date_published" : "2020-08-29T15:29:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/puasa-asyura/", "url" : "https://www.aantriono.com/en/puasa-asyura/", "summary": " IntroductionPuasa Asyura' In 1442 AH, the day of Ashura, 10 Muharram, fell on Saturday, August 29, 2020. Ashura holds a special place in the Islamic tradition, and the Prophet Muhammad gave particular attention to fasting on that day. Ibn Abbas narrated that he had never seen the Prophet give greater attention to fasting on any day outside Ramadan than he gave to the fast of Ashura.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5k-hTn90Wd1FbgWAGRc3t5sfgkAT5DmLZd8_OJ9ojcn4Zkj0EiDkAmySIzDIWwF8Flsw8T4dLjMf9syjrwhctfgMlr7zbc0-FjWOjKLc3a9Lmgo34a1bfNnRAwom7pScw_uJDi8dmdnk/s200/puasa.jpg\" title=\"Ashura Fast\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5k-hTn90Wd1FbgWAGRc3t5sfgkAT5DmLZd8_OJ9ojcn4Zkj0EiDkAmySIzDIWwF8Flsw8T4dLjMf9syjrwhctfgMlr7zbc0-FjWOjKLc3a9Lmgo34a1bfNnRAwom7pScw_uJDi8dmdnk/s200/puasa.jpg\" data-sub-html=\"\u003ch2\u003ePuasa Asyura\u0026#39;\u003c/h2\u003e\u003cp\u003eAshura Fast\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5k-hTn90Wd1FbgWAGRc3t5sfgkAT5DmLZd8_OJ9ojcn4Zkj0EiDkAmySIzDIWwF8Flsw8T4dLjMf9syjrwhctfgMlr7zbc0-FjWOjKLc3a9Lmgo34a1bfNnRAwom7pScw_uJDi8dmdnk/s200/puasa.jpg' alt=\"Ashura Fast\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003ePuasa Asyura'\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eIn 1442 AH, the day of Ashura, 10 Muharram, fell on Saturday, August 29, 2020. Ashura holds a special place in the Islamic tradition, and the Prophet Muhammad gave particular attention to fasting on that day. Ibn Abbas narrated that he had never seen the Prophet give greater attention to fasting on any day outside Ramadan than he gave to the fast of Ashura.\u003c/p\u003e\n\u003cp\u003eThe Prophet SAW also promised that fasting on the day of Ashura\u0026rsquo; could be a kaffarah (penance for sins) of the previous year. The scholars agree that fasting on the day of Ashura\u0026rsquo; is not obligatory; its ruling is sunnah.\u003c/p\u003e\n\u003ch2 id=\"stages-of-the-ashura-fast\" class=\"headerLink\"\u003e\n \u003ca href=\"#stages-of-the-ashura-fast\" class=\"header-mark\" aria-label=\"Direct link to section: Stages of the Ashura Fast\"\u003e\u003c/a\u003eStages of the Ashura Fast\u003c/h2\u003e\u003cp\u003eAl-Hafidz Ibn Rajab mentioned the stages of the Ashura fasting journey:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFirst\u003c/strong\u003e, the Prophet SAW had fasted on Ashura along with the polytheists of Mecca. \u0026lsquo;Aisyah RA narrated: \u0026ldquo;The Day of Ashura was a day of fasting for the Quraysh during the period of ignorance. And in the past, Rasulullah SAW also fasted on Ashura. When he arrived in Medina, he fasted, and ordered his companions to fast on Ashura.\u0026rdquo; (HR. Bukhari and Muslim)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSecond\u003c/strong\u003e, when the Prophet SAW arrived in Medina, he saw Jewish people fasting on Ashura. He then fasted and ordered his companions to fast as well. At that time, fasting on Ashura became an obligatory fast for Muslims.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThird\u003c/strong\u003e, when fasting in Ramadan was made obligatory, the ruling of fasting on Ashura\u0026rsquo; became recommended and no longer obligatory.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFourth\u003c/strong\u003e, at the end of the Prophet SAW\u0026rsquo;s life, he instructed his companions to fast on the 9th and 10th of Muharram, to differentiate it from the fasting of the Jews. Ibn Abbas RA narrated: \u0026ldquo;When the Prophet SAW carried out the fast of Ashura\u0026rsquo; and ordered his companions to fast, a companion said: \u0026lsquo;O Rasulullah, indeed the day of Ashura\u0026rsquo; is a day that is glorified by Jews and Christians.\u0026rsquo; Then the Prophet SAW said: \u0026lsquo;Next year, we will fast on the ninth day.\u0026rsquo; However, before the next year arrived, the Prophet SAW had already passed away.\u0026rdquo; (HR. Muslim)\u003c/p\u003e\n\u003cp\u003eMay Allah SWT lengthen our lives so that we can meet again in the month of Muharram next year and be given guidance and inayah to fast on the day of Ashura\u0026rsquo;. Aamiin\u0026hellip;\u003c/p\u003e", "language": "en" }, { "title" : "Solving Quadratic Equations Using the abc Formula", "date_published" : "2020-08-24T13:46:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/menyelesaikan-persamaan-kuadrat/", "url" : "https://www.aantriono.com/en/menyelesaikan-persamaan-kuadrat/", "summary": " IntroductionTo find the roots of a quadratic equation, we can use factoring, completing the square, or the quadratic formula often called the abc formula:\nx1,2=−b±b2−4ac2a x_{1,2}=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a} x1,2​=2a−b±b2−4ac​​This article focuses on solving quadratic equations using that formula.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003eTo find the roots of a quadratic equation, we can use factoring, completing the square, or the quadratic formula often called the \u003cstrong\u003eabc formula\u003c/strong\u003e:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx_{1,2}=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" 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nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThis article focuses on solving quadratic equations using that formula.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOEVfcypoIU61pCOMkzIHGxoNkBDJfIG7GJq3H6T3Y-f_cvhTGjrRnoly6HKn6-GB4AsJNIIjSE5VvPoec5lBQY4a_VKQGIJ_qkmh89oKNbjuZwGTEr9kEsOhUs3z-Tp3cJKuYm981KxE/w640-h367/abc.png' alt=\"Quadratic formula or abc formula\" \u003e\u003c/p\u003e\n\u003ch2 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example 1\"\u003e\u003c/a\u003eExample 1\u003c/h2\u003e\u003cp\u003eFind the roots of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2x^2+x-6=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"solution\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eFrom the equation, we get:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=-6\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eUse the abc formula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e48\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e49\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx_{1,2} \u0026amp;= \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\\\\n\u0026amp;= \\frac{-1 \\pm \\sqrt{1^2 - 4(2)(-6)}}{2(2)} \\\\\n\u0026amp;= \\frac{-1 \\pm \\sqrt{1+48}}{4} \\\\\n\u0026amp;= \\frac{-1 \\pm \\sqrt{49}}{4} \\\\\n\u0026amp;= \\frac{-1 \\pm 7}{4}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:12.8486em;vertical-align:-6.1743em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.6743em;\"\u003e\u003cspan style=\"top:-8.7139em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct mtight\"\u003e,\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.0979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3194em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.7491em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.5583em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.1743em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.6743em;\"\u003e\u003cspan style=\"top:-8.7139em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" 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style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.1743em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx_1 \u0026amp;= \\frac{-1+7}{4} = \\frac{6}{4} = \\frac{3}{2} \\\\\nx_2 \u0026amp;= \\frac{-1-7}{4} = \\frac{-8}{4} = -2\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan 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style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5574em;\"\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the roots are:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=\\frac{3}{2} \\quad \\text{and} \\quad x=-2\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example 2\"\u003e\u003c/a\u003eExample 2\u003c/h2\u003e\u003cp\u003eFind the roots of:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n2x^2+13x-7=0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.9474em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"solution-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#solution-1\" class=\"header-mark\" aria-label=\"Direct link to section: Solution\"\u003e\u003c/a\u003eSolution\u003c/h3\u003e\u003cp\u003eFrom the equation, we get:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea=2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003eb=13\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ec=-7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eUse the abc formula:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e169\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e56\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmsqrt\u003e\u003cmn\u003e225\u003c/mn\u003e\u003c/msqrt\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e±\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx_{1,2} \u0026amp;= \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\\\\n\u0026amp;= \\frac{-13 \\pm \\sqrt{13^2 - 4(2)(-7)}}{2(2)} \\\\\n\u0026amp;= \\frac{-13 \\pm \\sqrt{169+56}}{4} \\\\\n\u0026amp;= \\frac{-13 \\pm \\sqrt{225}}{4} \\\\\n\u0026amp;= \\frac{-13 \\pm 15}{4}\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:12.8486em;vertical-align:-6.1743em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.6743em;\"\u003e\u003cspan style=\"top:-8.7139em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003cspan class=\"mpunct mtight\"\u003e,\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.2861em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-6.0979em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.3194em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.7491em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:1.5583em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.1743em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.6743em;\"\u003e\u003cspan style=\"top:-8.7139em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.63em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.5904em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord sqrt\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.9134em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\" style=\"padding-left:0.833em;\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ec\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.8734em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e±\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:6.1743em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmsub\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msub\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e13\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e15\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e28\u003c/mn\u003e\u003c/mrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\nx_1 \u0026amp;= \\frac{-13+15}{4} = \\frac{2}{4} = \\frac{1}{2} \\\\\nx_2 \u0026amp;= \\frac{-13-15}{4} = \\frac{-28}{4} = -7\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.6149em;vertical-align:-2.0574em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5574em;\"\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.3011em;\"\u003e\u003cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.15em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5574em;\"\u003e\u003cspan style=\"top:-4.5574em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.25em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e13\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e15\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e28\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0574em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eTherefore, the roots are:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mfrac\u003e\u003cmspace width=\"1em\"/\u003e\u003cmtext\u003eand\u003c/mtext\u003e\u003cmspace width=\"1em\"/\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\nx=\\frac{1}{2} \\quad \\text{and} \\quad x=-7\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord text\"\u003e\u003cspan class=\"mord\"\u003eand\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:1em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ex\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThat is how quadratic equations can be solved using the quadratic formula or abc formula.\u003c/p\u003e\n", "language": "en" }, { "title" : "Indonesia Celebrates 75 Years", "date_published" : "2020-08-16T16:50:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/dirgahayu-indonesia-75-tahun/", "url" : "https://www.aantriono.com/en/dirgahayu-indonesia-75-tahun/", "summary": " Meaning of Indonesia\u0026rsquo;s 75th Independence Daydirgahayu indonesia 75 tahun Tomorrow, Monday, 17 August 2020, all Indonesian citizens will celebrate their 75th Independence Day. The theme of the Republic of Indonesia\u0026rsquo;s 75th anniversary is Advanced Indonesia and the logo used is Proudly Made in Indonesia. The choice of the theme \u0026ldquo;Advanced Indonesia\u0026rdquo; represents Pancasila as a guideline in national and state life in Indonesia.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"meaning-of-indonesias-75th-independence-day\" class=\"headerLink\"\u003e\n \u003ca href=\"#meaning-of-indonesias-75th-independence-day\" class=\"header-mark\" aria-label=\"Direct link to section: Meaning of Indonesia\u0026rsquo;s 75th Independence Day\"\u003e\u003c/a\u003eMeaning of Indonesia\u0026rsquo;s 75th Independence Day\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMxLmE47oL9iK1MQ2KKtkl90ayWGrTQvYrvR-RQyIk-r_pDAhoT5mhSkCZ4DsrYMAxDe4A6eCGg85nq1Qw5EUKBJgPrg_6oSXGzKZk1gLmx57Ml8ZBKYg5rLGUNyMHQhu8G6HKtBttu1Y/w400-h97/Layout\u0026#43;Spanduk\u0026#43;HUT\u0026#43;RI.jpg\" title=\"dirgahayu indonesia 75 tahun\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMxLmE47oL9iK1MQ2KKtkl90ayWGrTQvYrvR-RQyIk-r_pDAhoT5mhSkCZ4DsrYMAxDe4A6eCGg85nq1Qw5EUKBJgPrg_6oSXGzKZk1gLmx57Ml8ZBKYg5rLGUNyMHQhu8G6HKtBttu1Y/w400-h97/Layout\u0026#43;Spanduk\u0026#43;HUT\u0026#43;RI.jpg\" data-sub-html=\"\u003ch2\u003edirgahayu indonesia 75 tahun\u003c/h2\u003e\u003cp\u003edirgahayu indonesia 75 tahun\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMxLmE47oL9iK1MQ2KKtkl90ayWGrTQvYrvR-RQyIk-r_pDAhoT5mhSkCZ4DsrYMAxDe4A6eCGg85nq1Qw5EUKBJgPrg_6oSXGzKZk1gLmx57Ml8ZBKYg5rLGUNyMHQhu8G6HKtBttu1Y/w400-h97/Layout+Spanduk+HUT+RI.jpg' alt=\"dirgahayu indonesia 75 tahun\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003edirgahayu indonesia 75 tahun\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eTomorrow, Monday, 17 August 2020, all Indonesian citizens will celebrate their 75th Independence Day. The theme of the Republic of Indonesia\u0026rsquo;s 75th anniversary is \u003cstrong\u003eAdvanced Indonesia\u003c/strong\u003e and the logo used is \u003cstrong\u003eProudly Made in Indonesia\u003c/strong\u003e. The choice of the theme \u0026ldquo;Advanced Indonesia\u0026rdquo; represents Pancasila as a guideline in national and state life in Indonesia.\u003c/p\u003e\n\u003ch2 id=\"philosophy-of-the-advanced-indonesia-theme\" class=\"headerLink\"\u003e\n \u003ca href=\"#philosophy-of-the-advanced-indonesia-theme\" class=\"header-mark\" aria-label=\"Direct link to section: Philosophy of the \u0026ldquo;Advanced Indonesia\u0026rdquo; Theme\"\u003e\u003c/a\u003ePhilosophy of the \u0026ldquo;Advanced Indonesia\u0026rdquo; Theme\u003c/h2\u003e\u003cp\u003eThis can be seen from the number 75 in the Advanced Indonesia theme, which is inspired by the shield symbol in the Garuda Pancasila emblem. Apart from that, it also carries the meaning of equality and economic growth for the Indonesian people, and real progress in working to present the best results to all Indonesian people.\u003c/p\u003e\n\u003ch2 id=\"philosophy-of-the-proudly-made-in-indonesia-logo\" class=\"headerLink\"\u003e\n \u003ca href=\"#philosophy-of-the-proudly-made-in-indonesia-logo\" class=\"header-mark\" aria-label=\"Direct link to section: Philosophy of the \u0026ldquo;Proudly Made in Indonesia\u0026rdquo; Logo\"\u003e\u003c/a\u003ePhilosophy of the \u0026ldquo;Proudly Made in Indonesia\u0026rdquo; Logo\u003c/h2\u003e\u003cp\u003eMeanwhile, the meaning of the \u0026ldquo;Proudly Made in Indonesia\u0026rdquo; logo is an effort to foster Indonesian citizens\u0026rsquo; love for local or domestically made products.\u003c/p\u003e\n\u003cp\u003eHopefully in the future, Indonesia will become a country that is advanced in various aspects of life — sovereign in politics, self-reliant in the economy, and cultured in personality — becoming a just and prosperous country in accordance with the ideals of the \u003cem\u003efounding fathers\u003c/em\u003e of our beloved Republic of Indonesia.\u003c/p\u003e", "language": "en" }, { "title" : "Learning Through Blogs", "date_published" : "2020-08-15T14:00:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pembelajaran-melalui-blog/", "url" : "https://www.aantriono.com/en/pembelajaran-melalui-blog/", "summary": " Introduction In 2020, the world was shaken by the emergence of a mysterious virus known as COVID-19 (Corona Virus Disease 2019). The emergence of this virus was first discovered in the city of Wuhan, China at the end of December 2019. This virus spreads very quickly and has spread to almost all countries, including Indonesia, in just a few months.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTq6oc9cT5Qu2XtcBY8AYTb7IXPUWuBxmHXozCd3F5fBpdYc7bU3nPwnUnX8o-p9gzbVpdbrKk6YDEgIW2OHAG4j75L5VBWNbEPxkfVG7PvBzPzrqMGZ6dsrbbxoTD5B-izjaRnIrIMB4/s1600/blog.png\" style=\"margin-left: 1em; margin-right: 1em; text-align: justify;\"\u003e\u003cimg alt=\"pembelajaran melalui blog di masa pandemi\" border=\"0\" data-original-height=\"250\" data-original-width=\"250\" height=\"320\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTq6oc9cT5Qu2XtcBY8AYTb7IXPUWuBxmHXozCd3F5fBpdYc7bU3nPwnUnX8o-p9gzbVpdbrKk6YDEgIW2OHAG4j75L5VBWNbEPxkfVG7PvBzPzrqMGZ6dsrbbxoTD5B-izjaRnIrIMB4/w320-h320/blog.png\" title=\"pembelajaran melalui blog\" width=\"320\" /\u003e\u003c/a\u003e\nIn 2020, the world was shaken by the emergence of a mysterious virus known as COVID-19 (Corona Virus Disease 2019). The emergence of this virus was first discovered in the city of Wuhan, China at the end of December 2019. This virus spreads very quickly and has spread to almost all countries, including Indonesia, in just a few months.\u003c/p\u003e\n\u003ch2 id=\"changes-in-the-field-of-education\" class=\"headerLink\"\u003e\n \u003ca href=\"#changes-in-the-field-of-education\" class=\"header-mark\" aria-label=\"Direct link to section: Changes in the field of education\"\u003e\u003c/a\u003eChanges in the field of education\u003c/h2\u003e\u003cp\u003eVarious areas of life have felt the impact of this virus, including economics, industry, health and so on. The sector that is also affected by the emergence of this virus is the education sector. Ministries in various countries have taken steps in every school and university to conduct learning via the internet. Learning that originally used a face-to-face PBM system has changed to using an online learning system. The transition from face-to-face learning to online is certainly forcing various parties to be able to follow the process and flow, so that the learning system continues to run well. However, it turns out that this system is not working as effectively as imagined, in fact all parties are experiencing difficulties, not only students, parents, teachers and the government are also feeling it. This online learning has a huge impact, both positive and negative impacts. As we can see, not all of society is technologically literate, including teachers, students and parents who are still in the adaptation stage to current technological advances.\u003c/p\u003e\n\u003ch2 id=\"online-learning-support-application\" class=\"headerLink\"\u003e\n \u003ca href=\"#online-learning-support-application\" class=\"header-mark\" aria-label=\"Direct link to section: Online learning support application\"\u003e\u003c/a\u003eOnline learning support application\u003c/h2\u003e\u003cp\u003eVarious applications are used by teachers to support the implementation of online learning, some use \u003ca href=\"http://aantriono.gnomio.com/\"\u003e\u003cstrong\u003eMoodle LMS\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e, Schoology, Microsoft Teams, Google Classroom, \u003ca href=\"https://aframedia.igi.my.id/\"\u003e\u003cstrong\u003ePersonal Blog\u003c/strong\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e, and so forth. Of course, the use of this application is adjusted to your needs and readiness to utilize and use the application.\u003c/p\u003e\n\u003ch2 id=\"learning-using-blogs-and-moodle-lms\" class=\"headerLink\"\u003e\n \u003ca href=\"#learning-using-blogs-and-moodle-lms\" class=\"header-mark\" aria-label=\"Direct link to section: Learning using blogs and Moodle LMS\"\u003e\u003c/a\u003eLearning using blogs and Moodle LMS\u003c/h2\u003e\u003cp\u003eThe author himself, as a teacher, uses a personal blog and Moodle LMS to carry out online learning to students. The Moodle LMS that has been created is connected to the blog that is owned so that students do not have too much difficulty accessing the material and questions because it has been provided in the blog and Moodle itself.\u003c/p\u003e\n\u003ch2 id=\"benefits-of-learning-using-blogs\" class=\"headerLink\"\u003e\n \u003ca href=\"#benefits-of-learning-using-blogs\" class=\"header-mark\" aria-label=\"Direct link to section: Benefits of learning using blogs\"\u003e\u003c/a\u003eBenefits of learning using blogs\u003c/h2\u003e\u003cp\u003eThe benefits of blogs as a medium and learning resource include the following:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eBlog as a home for learning and sharing teachers. This means that teachers' creativity and passion can be channeled through blogs, such as creativity in writing and other works. Because print media is quite limited in containing our writings, blogs can be an outlet for writing creativity because they can be published directly without editing.\u003c/li\u003e\n\u003cli\u003eBlogs can lighten teachers' duties and burdens in teaching. Blogs will make it easier for teachers because everything can be included in the blog. For example, lesson materials, student assignments, student grade information. While students can easily download it without being limited by time and space. This method can certainly save time, energy and costs.\u003c/li\u003e\n\u003cli\u003e\nBlogs can increase students' interest in learning. With a blog, a teacher can post a problem or lesson material that is structured in formal but more relaxed language. Then students can blogwalk to the blog and teaching and learning activities can become more fun. Study material posted via blog media can become great content that is useful for the progress of the world of education.\n\u003c/li\u003e\n\u003cli\u003e\nBlogs can be accessed by anyone in the world. With blogs, teachers can share learning material not only with their students, but also with anyone who needs their ideas wherever they are, easily and cheaply.\n\u003c/li\u003e\n\u003cli\u003e\nBlogs can be a medium for friendship. Blogs can be used as a means to meet indirectly and communicate with each other because blogs can be accessed by anyone throughout the world.\n\u003c/li\u003e\n\u003c/ol\u003e\nFinally, considering the many benefits that can be obtained from using blogs as a medium and learning resource for teachers and students, teachers can certainly try to apply this media. Having a blog means having a virtual home. In this house, we can fill it with anything that is useful for other people, especially to support the teacher's task in educating the nation's generation.\n", "language": "en" }, { "title" : "Number Patterns and Sequences", "date_published" : "2020-07-21T01:48:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pola-dan-barisan-bilangan/", "url" : "https://www.aantriono.com/en/pola-dan-barisan-bilangan/", "summary": " Introduction to Number Patterns A number pattern means an arrangement of numbers that has a regular form or a number that is composed of several other numbers that form a pattern. And number patterns also have many types or types. On this occasion, we will study it together.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction-to-number-patterns\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction-to-number-patterns\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction to Number Patterns\"\u003e\u003c/a\u003eIntroduction to Number Patterns\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3rFtaByNW_nt8ClvSNJNoY9HcFlCAYZGcabDuVQ21S3yzkMmZoPL48S36AJyhfKsu0rfutOOjGspZuQBd1PlIBNWnbW8YolW7G-4OEvW5G59v2MVojLAVIMNG0uQOXKZERpFolbS3D2s/s1600/pola+bilangan.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"pola dan barisan bilangan\" border=\"0\" data-original-height=\"170\" data-original-width=\"297\" height=\"229\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3rFtaByNW_nt8ClvSNJNoY9HcFlCAYZGcabDuVQ21S3yzkMmZoPL48S36AJyhfKsu0rfutOOjGspZuQBd1PlIBNWnbW8YolW7G-4OEvW5G59v2MVojLAVIMNG0uQOXKZERpFolbS3D2s/w400-h229/pola+bilangan.jpg\" title=\"pola dan barisan bilangan\" width=\"400\" /\u003e\u003c/a\u003e\nA number pattern means an arrangement of numbers that has a regular form or a number that is composed of several other numbers that form a pattern. And number patterns also have many types or types. On this occasion, we will study it together.\u003c/p\u003e\n\u003ch2 id=\"types-of-number-patterns\" class=\"headerLink\"\u003e\n \u003ca href=\"#types-of-number-patterns\" class=\"header-mark\" aria-label=\"Direct link to section: Types of Number Patterns\"\u003e\u003c/a\u003eTypes of Number Patterns\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eOdd Number Patterns Odd number patterns are number patterns formed from odd numbers. Meanwhile, the definition of an odd number itself means a natural number that is not divisible by two or multiples.odd number patterns are: 1, 3, 5, 7, 9, . . . .Odd number pattern image:\u003ca href=\"https://rumusrumus.com/wp-content/uploads/2015/12/Pola-Bilangan.png\"\u003e\u003cimg src=\"https://rumusrumus.com/wp-content/uploads/2015/12/Pola-Bilangan-300x73.png\" /\u003e\u003c/a\u003e\nOdd Number Pattern Formula1 , 3 , 5 , 7 , . . . , n , then the formula for the nth odd number pattern is:Un = 2n – 1 Example :1 , 3 , 5 , 7 , . . . , to 10What is the pattern for the 10th odd number?Answer :Un = 2n – 1U10 = 2 . 10 – 1= 20 – 1 = 192. Even Number Patterns Even number patterns are number patterns formed from even numbers. Even numbers are natural numbers, namely natural numbers that are divisible by two or multiples thereof. The pattern of even numbers is: 2, 4, 6, 8, . . .Draw an even number pattern:\u003ca href=\"https://rumusrumus.com/wp-content/uploads/2015/12/pola.png\"\u003e\u003cimg src=\"https://rumusrumus.com/wp-content/uploads/2015/12/pola-300x64.png\" /\u003e\u003c/a\u003e\nEven number pattern formula2 , 4 , 6 , 8 , . . . . , n then the formula for the pattern of the nth even number is: Un = 2n Example :2 , 4 , 6 , 8 , . . . to 10. What is the pattern of even numbers to 10?answer :Un = 2nU10 = 2 x 10= 203. Square number pattern Square number pattern, namely a sequence of numbers that form a square pattern. The square number pattern is 1, 4, 9, 16, 25, . . .Square number pattern image:\u003ca href=\"https://rumusrumus.com/wp-content/uploads/2015/12/pol.png\"\u003e\u003cimg src=\"https://rumusrumus.com/wp-content/uploads/2015/12/pol.png\" /\u003e\u003c/a\u003e\nSquare number pattern formula1 , 4 , 9 , 16 , 25 , 36 , . . . , n maka rumus untuk mencari pola bilangan persegi ke n adalah :Un = n2From a sequence of numbers 1, 2, 9, 16, 25, 36, . . . , to 10 . What is the 10th number pattern in the square number pattern?Answer :Un = n2U10 = 102 = 1004. Rectangular Number Pattern A rectangular number pattern is a sequence of numbers that form a rectangular pattern. The rectangle pattern is 2 , 6 , 12 , 20 , 30 , . . .Rectangular Number Pattern Image:\u003ca href=\"https://rumusrumus.com/wp-content/uploads/2015/12/p.png\"\u003e\u003cimg src=\"https://rumusrumus.com/wp-content/uploads/2015/12/p-300x105.png\" /\u003e\u003c/a\u003e\nRectangular number pattern formula2 , 6 , 12 , 20 , 30 , . . . n, then the formula for the nth rectangular number pattern is:Un = n . n+1 Example :From a sequence of numbers 2, 6, 12, 20, 30, . . . , to 10 . What is the pattern for the 10th square number?Answer :Un = n . n+ 1U10 = 10 . 10+1= 10 . 11= 1105. Triangular Number Patterns A triangular number pattern is a sequence of numbers that form a triangular number pattern. The triangle number pattern is: 1, 3, 6, 10, 15, . . .Image of triangular number pattern:\u003ca href=\"https://rumusrumus.com/wp-content/uploads/2015/12/pl.png\"\u003e\u003cimg src=\"https://rumusrumus.com/wp-content/uploads/2015/12/pl.png\" /\u003e\u003c/a\u003e\nTriangular Number Pattern Formula:1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , . . . , to n . So the formula for the nth triangular number pattern is: Un = 1 / 2 n ( n + 1 ) Problems example :From a sequence of numbers 1, 3, 6, 10, 15, 21, 28, 36, . . . , to 10 . What is the 10th triangle number pattern?Answer :Un = 1/2 n ( n + 1 )U 10 = 1/2 .10 ( 10 + 1 )= 5 ( 11 ) = 556. FIBONACCI Number Pattern The Fibonacci number pattern is a number where each term is the sum of the two terms in front of it. Fibonacci number pattern:1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 56 , . . .2 , 2 , 4 , 6 , 10 , 16 , 26 , 42 , . . ..That is the basic explanation of number patterns in mathematics . In essence, a number pattern is a form of number sequence. When studied carefully, each pattern has its own structure and use, especially for understanding arithmetic and geometric series. I hope this is helpful.\u003c/li\u003e\n\u003c/ol\u003e", "language": "en" }, { "title" : "Exponents", "date_published" : "2020-07-13T14:17:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/bilangan-berpangkat/", "url" : "https://www.aantriono.com/en/bilangan-berpangkat/", "summary": " A. Introduction\nExponents are widely used in daily life, especially when writing very large or very small quantities. Learning exponents helps us write and calculate such quantities efficiently.\nB. Positive Integer Exponents DefinitionIf a∈Ra \\in \\mathbb{R}a∈R and nnn is a positive integer, then:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"a-introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#a-introduction\" class=\"header-mark\" aria-label=\"Direct link to section: A. Introduction\"\u003e\u003c/a\u003eA. Introduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRaO_077xvLPt5DLVhyh77eeRhzISLhaGBql1d9SD8tfejpwjIbSunDwAfGOPMT4q6rGyDZ_vZzYIKpgvOuqunUHPN0vXsZNHYK2FXkgYvVu4gEm_n4CPIOgDICT5F5R0lMYweCx3A_NE/w640-h636/Screenshot+2021-07-09+16.41.52.png' alt=\"Exponents\" \u003e\u003c/p\u003e\n\u003cp\u003eExponents are widely used in daily life, especially when writing very large or very small quantities. Learning exponents helps us write and calculate such quantities efficiently.\u003c/p\u003e\n\u003ch2 id=\"b-positive-integer-exponents\" class=\"headerLink\"\u003e\n \u003ca href=\"#b-positive-integer-exponents\" class=\"header-mark\" aria-label=\"Direct link to section: B. Positive Integer Exponents\"\u003e\u003c/a\u003eB. Positive Integer Exponents\u003c/h2\u003e\u003ch3 id=\"definition\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition\" class=\"header-mark\" aria-label=\"Direct link to section: Definition\"\u003e\u003c/a\u003eDefinition\u003c/h3\u003e\u003cp\u003eIf \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmi mathvariant=\"double-struck\"\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\in \\mathbb{R}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e∈\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6889em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathbb\"\u003eR\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a positive integer, then:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmunder\u003e\u003cmunder\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo\u003e⋯\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"true\"\u003e⏟\u003c/mo\u003e\u003c/munder\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmtext\u003e times\u003c/mtext\u003e\u003c/mrow\u003e\u003c/munder\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^n=\\underbrace{a \\times a \\times \\cdots \\times a}_{n\\ \\text{times}}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7144em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.9822em;vertical-align:-1.3988em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord munder\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5833em;\"\u003e\u003cspan style=\"top:-1.6012em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003cspan class=\"mspace mtight\"\u003e\u003cspan class=\"mtight\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord text mtight\"\u003e\u003cspan class=\"mord mtight\"\u003etimes\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord munder\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5833em;\"\u003e\u003cspan class=\"svg-align\" style=\"top:-2.2687em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"stretchy\" style=\"height:0.548em;min-width:1.6em;\"\u003e\u003cspan class=\"brace-left\" style=\"height:0.548em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"0.548em\" viewBox=\"0 0 400000 548\" preserveAspectRatio=\"xMinYMin slice\"\u003e\u003cpath d=\"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003cspan class=\"brace-center\" style=\"height:0.548em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"0.548em\" viewBox=\"0 0 400000 548\" preserveAspectRatio=\"xMidYMin slice\"\u003e\u003cpath d=\"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003cspan class=\"brace-right\" style=\"height:0.548em;\"\u003e\u003csvg xmlns=\"http://www.w3.org/2000/svg\" width=\"400em\" height=\"0.548em\" viewBox=\"0 0 400000 548\" preserveAspectRatio=\"xMaxYMin slice\"\u003e\u003cpath d=\"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z\"/\u003e\u003c/svg\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e⋯\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7313em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3988em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea^n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6644em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6644em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is called a power,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the base,\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is the exponent.\u003c/li\u003e\n\u003c/ul\u003e\n\u003ch3 id=\"example\" class=\"headerLink\"\u003e\n \u003ca href=\"#example\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eWrite the following powers as repeated multiplication, then calculate the result:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2^5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-3)^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0.5\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0.5)^4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-4)^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e32\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e2^5=2 \\times 2 \\times 2 \\times 2 \\times 2=32\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e32\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-3)^2=(-3)\\times(-3)=9\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0.5\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0.5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0.5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0.5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e0.5\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0.0625\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(0.5)^4=(0.5)\\times(0.5)\\times(0.5)\\times(0.5)=0.0625\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0.0625\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e64\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-4)^3=(-4)\\times(-4)\\times(-4)=-64\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e64\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"c-properties-of-exponent-operations\" class=\"headerLink\"\u003e\n \u003ca href=\"#c-properties-of-exponent-operations\" class=\"header-mark\" aria-label=\"Direct link to section: C. Properties of Exponent Operations\"\u003e\u003c/a\u003eC. Properties of Exponent Operations\u003c/h2\u003e\u003ch3 id=\"1-multiplication-property\" class=\"headerLink\"\u003e\n \u003ca href=\"#1-multiplication-property\" class=\"header-mark\" aria-label=\"Direct link to section: 1. Multiplication Property\"\u003e\u003c/a\u003e1. Multiplication Property\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^m \\times a^n = a^{m+n}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7977em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7144em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8213em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8213em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewhere \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e is a real number and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo separator=\"true\"\u003e,\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em,n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mpunct\"\u003e,\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.1667em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e are positive integers.\u003c/p\u003e\n\u003ch4 id=\"example-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-1\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003cp\u003eSimplify:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6^3 \\times 6^4\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-4)\\times(-4)^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e5^2\\times 3^3\\times 2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7a^3\\times b^4 \\times 3a^2 \\times b\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-1\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e6^3 \\times 6^4 = 6^{3+4}=6^7\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-4)\\times(-4)^2 = (-4)^{1+2}=(-4)^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e4\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eSince the bases are not the same, the expression cannot be simplified further.\u003c/li\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e21\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n7a^3\\times b^4 \\times 3a^2 \\times b\n\u0026amp;= 7a^3\\times 3a^2\\times b^4\\times b \\\\\n\u0026amp;= 21a^{3+2}b^{4+1} \\\\\n\u0026amp;= 21a^5b^5\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5723em;vertical-align:-2.0362em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e21\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"2-division-property\" class=\"headerLink\"\u003e\n \u003ca href=\"#2-division-property\" class=\"header-mark\" aria-label=\"Direct link to section: 2. Division Property\"\u003e\u003c/a\u003e2. Division Property\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{a^m}{a^n}=a^{m-n}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0274em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3414em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5904em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.6644em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8213em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8213em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\neq 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em\u0026gt;n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u0026gt;\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch4 id=\"example-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-2\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{6^{12}}{6^{10}}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{9^3}{6^2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e9\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{(-3)^4\\times (-3)^3}{(-3)^2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4271em;vertical-align:-0.936em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-2\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e12\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e10\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{6^{12}}{6^{10}}=6^{12-10}=6^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e12\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e12\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e10\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e6\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003eSince the bases are different, the expression cannot be simplified further.\u003c/li\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{(-3)^4\\times (-3)^3}{(-3)^2}\n=\\frac{(-3)^{4+3}}{(-3)^2}\n=\\frac{(-3)^7}{(-3)^2}\n=(-3)^{7-2}\n=(-3)^5\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4271em;vertical-align:-0.936em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4271em;vertical-align:-0.936em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.4271em;vertical-align:-0.936em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.1141em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"3-power-of-a-power\" class=\"headerLink\"\u003e\n \u003ca href=\"#3-power-of-a-power\" class=\"header-mark\" aria-label=\"Direct link to section: 3. Power of a Power\"\u003e\u003c/a\u003e3. Power of a Power\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n(a^m)^n=a^{m\\times n}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8213em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8213em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e×\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-3\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(5^4)^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{2^5\\times (2^3)^2}{2^4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan 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style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-3\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-3\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(5^4)^2=5^{4\\times 2}=5^8\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e×\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e11\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e11\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n\\frac{2^5\\times (2^3)^2}{2^4}\n\u0026amp;= \\frac{2^5\\times 2^6}{2^4} 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style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.0127em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan 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style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-5.0127em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan 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style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-2.5356em;\"\u003e\u003cspan class=\"pstrut\" 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style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-0.6854em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e11\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:0.8387em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.4911em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:4.9898em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"4-addition-and-subtraction-property\" class=\"headerLink\"\u003e\n \u003ca href=\"#4-addition-and-subtraction-property\" class=\"header-mark\" aria-label=\"Direct link to section: 4. Addition and Subtraction Property\"\u003e\u003c/a\u003e4. Addition and Subtraction Property\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^n+a^m=a^n(1+a^{m-n})\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7977em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7144em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0713em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8213em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003efor \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e≥\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003em \\ge n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7719em;vertical-align:-0.136em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003em\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e≥\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^n-a^m=a^n(1-a^{m-n})\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7977em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7144em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0713em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8213em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eor:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003em\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^m-a^n=a^n(a^{m-n}-1)\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7977em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.7144em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0713em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7144em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8213em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003em\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch4 id=\"example-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-4\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-8)^3+(-8)^5\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e+\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e7^7-7^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8974em;vertical-align:-0.0833em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-4\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-4\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\"\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n(-8)^3+(-8)^5\n\u0026amp;= (-8)^3+(-8)^{3+2} \\\\\n\u0026amp;= (-8)^3+(-8)^3\\cdot(-8)^2 \\\\\n\u0026amp;= (-8)^3(1+(-8)^2)\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan 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displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmstyle scriptlevel=\"0\" displaystyle=\"true\"\u003e\u003cmrow\u003e\u003cmrow\u003e\u003c/mrow\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\begin{aligned}\n7^7-7^3\n\u0026amp;= 7^{4+3}-7^3 \\\\\n\u0026amp;= 7^4\\cdot 7^3-7^3 \\\\\n\u0026amp;= 7^3(7^4-1)\n\\end{aligned}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:4.5723em;vertical-align:-2.0362em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mtable\"\u003e\u003cspan class=\"col-align-r\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"col-align-l\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.5362em;\"\u003e\u003cspan style=\"top:-4.6721em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.1479em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-1.6238em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e−\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.0362em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"d-negative-integer-exponents\" class=\"headerLink\"\u003e\n \u003ca href=\"#d-negative-integer-exponents\" class=\"header-mark\" aria-label=\"Direct link to section: D. Negative Integer Exponents\"\u003e\u003c/a\u003eD. Negative Integer Exponents\u003c/h2\u003e\u003cp\u003eObserve:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{2^2}{2^4}\n=\\frac{2\\times2}{2\\times2\\times2\\times2}\n=\\frac{1}{2\\times2}\n=\\frac{1}{2^2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eAlso:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{2^2}{2^4}=2^{2-4}=2^{-2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eSo:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{1}{2^2}=2^{-2}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"definition-1\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-1\" class=\"header-mark\" aria-label=\"Direct link to section: Definition\"\u003e\u003c/a\u003eDefinition\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^{-n}=\\frac{1}{a^n}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8213em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8213em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.5904em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mathnormal mtight\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\neq 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003en\u003c/mi\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003en\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.4306em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003en\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e a positive integer.\u003c/p\u003e\n\u003ch3 id=\"example-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-5\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003col\u003e\n\u003cli\u003eRewrite in positive-exponent form:\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^{-5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-8)^{-4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea^{-2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003eRewrite in negative-exponent form:\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{7^2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{2^6}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{a^9}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-5\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-5\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003ePositive-exponent form:\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e3^{-5}=\\dfrac{1}{3^5}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(-8)^{-4}=\\dfrac{1}{(-8)^4}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.2574em;vertical-align:-0.936em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e−\u003c/span\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e4\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.936em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea^{-2}=\\dfrac{1}{a^2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003cli\u003eNegative-exponent form:\n\u003cul\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{7^2}=7^{-2}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{2^6}=2^{-6}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\\dfrac{1}{a^9}=a^{-9}\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e9\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"e-zero-exponent\" class=\"headerLink\"\u003e\n \u003ca href=\"#e-zero-exponent\" class=\"header-mark\" aria-label=\"Direct link to section: E. Zero Exponent\"\u003e\u003c/a\u003eE. Zero Exponent\u003c/h2\u003e\u003cp\u003eObserve:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{3^2}{3^2}=\\frac{3\\times3}{3\\times3}=1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7693em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eand:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{3^2}{3^2}=3^{2-2}=3^0\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003eThus:\u003c/p\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n3^0=1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch3 id=\"definition-2\" class=\"headerLink\"\u003e\n \u003ca href=\"#definition-2\" class=\"header-mark\" aria-label=\"Direct link to section: Definition\"\u003e\u003c/a\u003eDefinition\u003c/h3\u003e\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\na^0=1\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8641em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8641em;\"\u003e\u003cspan style=\"top:-3.113em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cp\u003ewith \u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmo mathvariant=\"normal\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003ea \\neq 0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mrel\"\u003e\u003cspan class=\"mord vbox\"\u003e\u003cspan class=\"thinbox\"\u003e\u003cspan class=\"rlap\"\u003e\u003cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"\u003e\u003c/span\u003e\u003cspan class=\"inner\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mrel\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"fix\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003ch3 id=\"example-6\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-6\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003cp\u003eCalculate:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(5)^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(12)^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e34\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e34a^2b^0\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e34\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-6\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-6\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(5)^0=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e5\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e12\u003c/mn\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(12)^0=1\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e12\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmn\u003e34\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e34\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e⋅\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e34\u003c/mn\u003e\u003cmsup\u003e\u003cmi\u003ea\u003c/mi\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e34a^2b^0=34a^2\\cdot 1=34a^2\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e34\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003eb\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e0\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e34\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e⋅\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.6444em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e1\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:0.8141em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e34\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord mathnormal\"\u003ea\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"f-fractional-bases-with-integer-exponents\" class=\"headerLink\"\u003e\n \u003ca href=\"#f-fractional-bases-with-integer-exponents\" class=\"header-mark\" aria-label=\"Direct link to section: F. Fractional Bases with Integer Exponents\"\u003e\u003c/a\u003eF. Fractional Bases with Integer Exponents\u003c/h2\u003e\u003cp\u003eNumbers that can be written as fractions are called rational numbers. The properties of exponents also apply to rational numbers.\u003c/p\u003e\n\u003ch3 id=\"example-7\" class=\"headerLink\"\u003e\n \u003ca href=\"#example-7\" class=\"header-mark\" aria-label=\"Direct link to section: Example\"\u003e\u003c/a\u003eExample\u003c/h3\u003e\u003col\u003e\n\u003cli\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmstyle displaystyle=\"true\" scriptlevel=\"0\"\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mstyle\u003e\u003cmsup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e(\\dfrac{2}{3})^3\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord 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class=\"strut\" style=\"height:4.2949em;vertical-align:-1.8974em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:2.3974em;\"\u003e\u003cspan style=\"top:-2.11em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.5514em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-4.3974em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.3214em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan 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style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mopen\"\u003e(\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose\"\u003e\u003cspan class=\"mclose\"\u003e)\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.8974em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch4 id=\"answer-7\" class=\"headerLink\"\u003e\n \u003ca href=\"#answer-7\" class=\"header-mark\" aria-label=\"Direct link to section: Answer\"\u003e\u003c/a\u003eAnswer\u003c/h4\u003e\u003col\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmn\u003e27\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\left(\\frac{2}{3}\\right)^3\n=\\frac{2}{3}\\times\\frac{2}{3}\\times\\frac{2}{3}\n=\\frac{2^3}{3^3}\n=\\frac{8}{27}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.604em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose 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style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.1771em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.4911em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e3\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.7401em;\"\u003e\u003cspan style=\"top:-2.989em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8141em;\"\u003e\u003cspan style=\"top:-3.063em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e3\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e27\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e8\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003col start=\"2\"\u003e\n\u003cli\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cspan class=\"katex-display\"\u003e\u003cspan class=\"katex\"\u003e\u003cspan class=\"katex-mathml\"\u003e\u003cmath xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003csemantics\u003e\u003cmrow\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e×\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003c/mrow\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/msup\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/msup\u003e\u003c/mfrac\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo fence=\"true\"\u003e(\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003cmo fence=\"true\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmfrac\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mfrac\u003e\u003c/mrow\u003e\u003cannotation encoding=\"application/x-tex\"\u003e\n\\frac{\\left(\\frac{2}{7}\\right)^5 \\times \\left(\\frac{2}{7}\\right)^2}{\\left(\\frac{2}{7}\\right)^6}\n=\\frac{\\left(\\frac{2}{7}\\right)^{5+2}}{\\left(\\frac{2}{7}\\right)^6}\n=\\frac{\\left(\\frac{2}{7}\\right)^7}{\\left(\\frac{2}{7}\\right)^6}\n=\\left(\\frac{2}{7}\\right)^{7-6}\n=\\frac{2}{7}\n\u003c/annotation\u003e\u003c/semantics\u003e\u003c/math\u003e\u003c/span\u003e\u003cspan class=\"katex-html\" aria-hidden=\"true\"\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.088em;vertical-align:-1.294em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.794em;\"\u003e\u003cspan style=\"top:-2.11em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.054em;\"\u003e\u003cspan style=\"top:-3.3029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.284em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.794em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.054em;\"\u003e\u003cspan style=\"top:-3.3029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"mbin\"\u003e×\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2222em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.054em;\"\u003e\u003cspan style=\"top:-3.3029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.294em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.088em;vertical-align:-1.294em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.794em;\"\u003e\u003cspan style=\"top:-2.11em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.054em;\"\u003e\u003cspan style=\"top:-3.3029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.284em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.794em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.054em;\"\u003e\u003cspan style=\"top:-3.3029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e5\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e+\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.294em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:3.088em;vertical-align:-1.294em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.794em;\"\u003e\u003cspan style=\"top:-2.11em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.054em;\"\u003e\u003cspan style=\"top:-3.3029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.284em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.794em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3.054em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.8451em;\"\u003e\u003cspan style=\"top:-2.655em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.394em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.345em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size1\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.054em;\"\u003e\u003cspan style=\"top:-3.3029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.294em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.604em;vertical-align:-0.95em;\"\u003e\u003c/span\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"minner\"\u003e\u003cspan class=\"mopen delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e(\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose delimcenter\" style=\"top:0em;\"\u003e\u003cspan class=\"delimsizing size3\"\u003e)\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"msupsub\"\u003e\u003cspan class=\"vlist-t\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.654em;\"\u003e\u003cspan style=\"top:-3.9029em;margin-right:0.05em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:2.7em;\"\u003e\u003c/span\u003e\u003cspan class=\"sizing reset-size6 size3 mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e\u003cspan class=\"mord mtight\"\u003e7\u003c/span\u003e\u003cspan class=\"mbin mtight\"\u003e−\u003c/span\u003e\u003cspan class=\"mord mtight\"\u003e6\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003cspan class=\"mrel\"\u003e=\u003c/span\u003e\u003cspan class=\"mspace\" style=\"margin-right:0.2778em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"base\"\u003e\u003cspan class=\"strut\" style=\"height:2.0074em;vertical-align:-0.686em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mopen nulldelimiter\"\u003e\u003c/span\u003e\u003cspan class=\"mfrac\"\u003e\u003cspan class=\"vlist-t vlist-t2\"\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:1.3214em;\"\u003e\u003cspan style=\"top:-2.314em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e7\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.23em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\"\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"top:-3.677em;\"\u003e\u003cspan class=\"pstrut\" style=\"height:3em;\"\u003e\u003c/span\u003e\u003cspan class=\"mord\"\u003e\u003cspan class=\"mord\"\u003e2\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-s\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"vlist-r\"\u003e\u003cspan class=\"vlist\" style=\"height:0.686em;\"\u003e\u003cspan\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"mclose nulldelimiter\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003ch2 id=\"g-evaluation\" class=\"headerLink\"\u003e\n \u003ca href=\"#g-evaluation\" class=\"header-mark\" aria-label=\"Direct link to section: G. Evaluation\"\u003e\u003c/a\u003eG. Evaluation\u003c/h2\u003e\u003cp\u003eTo measure your understanding of exponents, follow this evaluation:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003echoose the correct answer,\u003c/li\u003e\n\u003cli\u003eclick \u003ccode\u003eNext\u003c/code\u003e to move to the next question,\u003c/li\u003e\n\u003cli\u003eclick \u003ccode\u003eSubmit\u003c/code\u003e to send your answers,\u003c/li\u003e\n\u003cli\u003echeck the score shown below the questions,\u003c/li\u003e\n\u003cli\u003ethe final value is the score divided by the maximum score, multiplied by 100,\u003c/li\u003e\n\u003cli\u003ethe minimum passing score is 70,\u003c/li\u003e\n\u003cli\u003ethe time limit is 30 minutes.\u003c/li\u003e\n\u003c/ol\u003e\n\u003ch2 id=\"h-closing\" class=\"headerLink\"\u003e\n \u003ca href=\"#h-closing\" class=\"header-mark\" aria-label=\"Direct link to section: H. Closing\"\u003e\u003c/a\u003eH. Closing\u003c/h2\u003e\u003cp\u003eThat concludes this material on exponents for Grade IX.\u003c/p\u003e\n", "language": "en" }, { "title" : "Behind the Blessings We Receive", "date_published" : "2020-07-13T03:06:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/dibalik-rezeki-kita/", "url" : "https://www.aantriono.com/en/dibalik-rezeki-kita/", "summary": " Reflections on Sustenancehuman sustenance Every blessing we receive is often connected to someone else\u0026rsquo;s livelihood. When we spend money at the market, for example, part of that blessing becomes the means by which a trader provides for his family.\nBehind a large shopping trip, there may be a merchant hoping to pay a child\u0026rsquo;s school fees. Behind an ordinary transaction, there may be another household quietly sustained by it.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"reflections-on-sustenance\" class=\"headerLink\"\u003e\n \u003ca href=\"#reflections-on-sustenance\" class=\"header-mark\" aria-label=\"Direct link to section: Reflections on Sustenance\"\u003e\u003c/a\u003eReflections on Sustenance\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB2tLJMlfpkU0UAB2nkyk1eMxVaR8ptZ4hbMMVNUBEYQAm9W2e6_dO2zOfNKa5QjCmG-QWbSHZA2uxP5eO7J50Ok_H7-EpYKKcfTFACRIGjylFzcg_ihsD9-0x759F1lnt76dSa9pQqyk/w200-h200/rezeki.jpg\" title=\"human sustenance\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB2tLJMlfpkU0UAB2nkyk1eMxVaR8ptZ4hbMMVNUBEYQAm9W2e6_dO2zOfNKa5QjCmG-QWbSHZA2uxP5eO7J50Ok_H7-EpYKKcfTFACRIGjylFzcg_ihsD9-0x759F1lnt76dSa9pQqyk/w200-h200/rezeki.jpg\" data-sub-html=\"\u003ch2\u003ehuman sustenance\u003c/h2\u003e\u003cp\u003ehuman sustenance\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB2tLJMlfpkU0UAB2nkyk1eMxVaR8ptZ4hbMMVNUBEYQAm9W2e6_dO2zOfNKa5QjCmG-QWbSHZA2uxP5eO7J50Ok_H7-EpYKKcfTFACRIGjylFzcg_ihsD9-0x759F1lnt76dSa9pQqyk/w200-h200/rezeki.jpg' alt=\"human sustenance\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003ehuman sustenance\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eEvery blessing we receive is often connected to someone else\u0026rsquo;s livelihood. When we spend money at the market, for example, part of that blessing becomes the means by which a trader provides for his family.\u003c/p\u003e\n\u003cp\u003eBehind a large shopping trip, there may be a merchant hoping to pay a child\u0026rsquo;s school fees. Behind an ordinary transaction, there may be another household quietly sustained by it.\u003c/p\u003e\n\u003cp\u003eBehind a damaged motorbike or car, there is a repairman\u0026rsquo;s fortune to buy food for his family.\u003c/p\u003e\n\u003cp\u003eBehind the ATM with the balance, there are the salaries of bank employees, who always smile to serve us when we go to the bank.\u003c/p\u003e\n\u003cp\u003eBehind our wallets when we pay at the supermarket, there is the fortune of an employee who is waiting to be paid to buy milk for his child.\u003c/p\u003e\n\u003cp\u003eBehind our fortune in buying ready-to-eat food to satisfy our hunger, there is the fortune of those who hope for it every day.\u003c/p\u003e\n\u003cp\u003eBehind our sustenance to repair the house to make it better, there is the sustenance of the builders to meet the needs of their children and wives at home.\u003c/p\u003e\n\u003cp\u003eBehind our fortune in repairing a punctured tire, there is the fortune of a patchworker who always hopes and looks for halal sustenance.\u003c/p\u003e\n\u003ch2 id=\"interconnection-of-human-livelihoods\" class=\"headerLink\"\u003e\n \u003ca href=\"#interconnection-of-human-livelihoods\" class=\"header-mark\" aria-label=\"Direct link to section: Interconnection of Human Livelihoods\"\u003e\u003c/a\u003eInterconnection of Human Livelihoods\u003c/h2\u003e\u003cp\u003eBehind our sustenance is someone else\u0026rsquo;s sustenance. Not because of our great efforts or abilities, but Allah\u0026rsquo;s way of providing sustenance is extraordinary and will definitely not be out of reach from our minds.\u003c/p\u003e\n\u003cp\u003eWhat we enjoy should be our sustenance, and what we give is the sustenance of other people that Allah has entrusted through us.\u003c/p\u003e\n\u003cp\u003eLet us not be sad when something happens and instead spend a little more of the sustenance that Allah has given us. Widen and broaden our minds and hearts, knowing that it is someone else\u0026rsquo;s sustenance that Allah has entrusted to us.\u003c/p\u003e\n\u003cp\u003eSource: Telegram Group\u003c/p\u003e", "language": "en" }, { "title" : "Every Episode Matters", "date_published" : "2020-07-13T02:44:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/setiap-episode-itu-penting/", "url" : "https://www.aantriono.com/en/setiap-episode-itu-penting/", "summary": " Every Episode of Life Mattersepisodes of life These days many people eagerly follow Korean dramas or long-running television series, never wanting to miss a single episode. Every episode matters because each one carries a piece of the story. Miss one, and the narrative starts to feel incomplete.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"every-episode-of-life-matters\" class=\"headerLink\"\u003e\n \u003ca href=\"#every-episode-of-life-matters\" class=\"header-mark\" aria-label=\"Direct link to section: Every Episode of Life Matters\"\u003e\u003c/a\u003eEvery Episode of Life Matters\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja42u2_L-6upXa-gVYJd6GrWPtlyk2n8g-WqYbMDlZIA0eWUeXBqRdHfFN2x20_8AzpOFtYYnIYOF8Kx5rIs0641pk2MmSyRi6XmmzCud0kCpRHXe1e564ksLKF2SLP4HZlhnitb0g6Io/w400-h210/episode.jpg\" title=\"episodes of human life\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja42u2_L-6upXa-gVYJd6GrWPtlyk2n8g-WqYbMDlZIA0eWUeXBqRdHfFN2x20_8AzpOFtYYnIYOF8Kx5rIs0641pk2MmSyRi6XmmzCud0kCpRHXe1e564ksLKF2SLP4HZlhnitb0g6Io/w400-h210/episode.jpg\" data-sub-html=\"\u003ch2\u003eepisodes of life\u003c/h2\u003e\u003cp\u003eepisodes of human life\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja42u2_L-6upXa-gVYJd6GrWPtlyk2n8g-WqYbMDlZIA0eWUeXBqRdHfFN2x20_8AzpOFtYYnIYOF8Kx5rIs0641pk2MmSyRi6XmmzCud0kCpRHXe1e564ksLKF2SLP4HZlhnitb0g6Io/w400-h210/episode.jpg' alt=\"episodes of human life\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003eepisodes of life\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003eThese days many people eagerly follow Korean dramas or long-running television series, never wanting to miss a single episode. Every episode matters because each one carries a piece of the story. Miss one, and the narrative starts to feel incomplete.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003ch2 id=\"valuing-each-day-as-a-new-chapter\" class=\"headerLink\"\u003e\n \u003ca href=\"#valuing-each-day-as-a-new-chapter\" class=\"header-mark\" aria-label=\"Direct link to section: Valuing Each Day as a New Chapter\"\u003e\u003c/a\u003eValuing Each Day as a New Chapter\u003c/h2\u003e\u003cp\u003eLife works in much the same way. Each day is like an episode: important, connected, and worth living well. That is why every day should be filled with goodness, so that the next chapter of our lives becomes something meaningful — not only for ourselves, but also for the people around us.\u003c/p\u003e\n\u003cp\u003eWhen we live each day with purpose and kindness, the episodes that follow become something others look forward to. Who waits for them? Our family, our friends, our community, and Allah — because every act of goodness we do is something He takes pleasure in when it is done sincerely for His sake.\u003c/p\u003e\n\u003cp\u003eSource: Telegram Group\u003c/p\u003e", "language": "en" }, { "title" : "This Too Shall Pass", "date_published" : "2020-07-09T06:10:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/badai-pasti-berlalu/", "url" : "https://www.aantriono.com/en/badai-pasti-berlalu/", "summary": " Introductiontyphoon Peace be upon you\u0026hellip; Friends, on this occasion Afra Media would like to share a short lesson through the following story. May it be of benefit.\nDON\u0026rsquo;T STOP IN THE MIDDLE OF A STORMA young man was driving with his mother. After they had traveled several dozen kilometers, dark clouds gathered and strong winds began to blow. The sky grew dim, and some vehicles started pulling over to the side of the road.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOJHvd2fOD4mUPT0NnKdtENTEc5kl0Q8LeSiQekeNUGIbKZGhp2L0hrFDfhpW8Tfl8fx9xtlAKL1-RIlQBQCqURFlbR4Rw6Vtrpd94gaOlcaQUow6IH4OBMTk-k4W99VB9EVOjCmkXW3k/w400-h222/badai.jpg\" title=\"typhoon in Indonesia\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOJHvd2fOD4mUPT0NnKdtENTEc5kl0Q8LeSiQekeNUGIbKZGhp2L0hrFDfhpW8Tfl8fx9xtlAKL1-RIlQBQCqURFlbR4Rw6Vtrpd94gaOlcaQUow6IH4OBMTk-k4W99VB9EVOjCmkXW3k/w400-h222/badai.jpg\" data-sub-html=\"\u003ch2\u003etyphoon\u003c/h2\u003e\u003cp\u003etyphoon in Indonesia\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOJHvd2fOD4mUPT0NnKdtENTEc5kl0Q8LeSiQekeNUGIbKZGhp2L0hrFDfhpW8Tfl8fx9xtlAKL1-RIlQBQCqURFlbR4Rw6Vtrpd94gaOlcaQUow6IH4OBMTk-k4W99VB9EVOjCmkXW3k/w400-h222/badai.jpg' alt=\"typhoon in Indonesia\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003etyphoon\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003ePeace be upon you\u0026hellip; Friends, on this occasion Afra Media would like to share a short lesson through the following story. May it be of benefit.\u003c/p\u003e\n\u003ch2 id=\"dont-stop-in-the-middle-of-a-storm\" class=\"headerLink\"\u003e\n \u003ca href=\"#dont-stop-in-the-middle-of-a-storm\" class=\"header-mark\" aria-label=\"Direct link to section: DON\u0026rsquo;T STOP IN THE MIDDLE OF A STORM\"\u003e\u003c/a\u003eDON\u0026rsquo;T STOP IN THE MIDDLE OF A STORM\u003c/h2\u003e\u003cp\u003eA young man was driving with his mother. After they had traveled several dozen kilometers, dark clouds gathered and strong winds began to blow. The sky grew dim, and some vehicles started pulling over to the side of the road.\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Mom, should we stop?\u0026rdquo; the son asked.\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Keep going,\u0026rdquo; his mother replied.\u003c/p\u003e\n\u003cp\u003eSo he continued driving. The sky became darker, the wind grew stronger, and heavy rain began to fall. Some trees had fallen, and others were bending hard in the wind. The situation was frightening. Even large vehicles began to stop by the roadside.\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Mom\u0026hellip;?\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Keep driving,\u0026rdquo; she said again, still looking straight ahead.\u003c/p\u003e\n\u003cp\u003eThe son pressed on with difficulty. The rain was so heavy that he could see only a few meters in front of him. He began to feel afraid, but he kept driving, even if only very slowly.\u003c/p\u003e\n\u003cp\u003eAfter a few more kilometers, the rain began to ease and the wind slowly calmed down. A little farther on, they finally reached a dry area where the sun was shining.\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Now stop the car and step outside,\u0026rdquo; his mother said.\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Why now?\u0026rdquo; he asked.\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;So you can see what would have happened if we had stopped in the middle of the storm.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eHe got out of the car and looked back. In the distance, the storm was still raging. He imagined the people who had remained trapped there.\u003c/p\u003e\n\u003cp\u003eAt that moment he understood: never stop in the middle of a storm, because that is where fear keeps you stuck.\u003c/p\u003e\n\u003cp\u003eWhen we face a storm in life, we need to keep moving. Do not stop in despair, because despair only keeps us trapped in what frightens us. Do what you can, keep trusting, and remember that every storm will pass.\u003c/p\u003e\n\u003cp\u003eKeep moving forward, because beyond hardship there is clarity, and with patience there is a way through.\u003c/p\u003e\n\u003cp\u003eLife does not always run smoothly. We need stones in the road so that we learn to walk carefully. We need thorny paths so that we learn vigilance. We need crossroads so that we become wise in choosing. We need direction so that we can keep hope for the future. We need problems so that we recognize the strength God has placed within us. We need sacrifice so that we learn how to strive with sincerity and intelligence. We need tears so that we grow humble. We need criticism so that we learn to value correction. We need laughter and smiles so that we remember gratitude. We need other people so that we remember we are not alone.\u003c/p\u003e\n\u003cp\u003eDo not answer problems with complaint and anger. Face them with patience, gratitude, and a heart that refuses to give up.\u003c/p\u003e\n\u003cp\u003eKeep moving even when obstacles come. Do not be afraid. When there is no wall to lean on, there is still a floor on which to bow.\u003c/p\u003e\n\u003cp\u003eThe finest good deeds are often the ones unseen by others, yet deeply felt in the heart.\u003c/p\u003e\n\u003cp\u003eDo not keep counting what is gone; count what still remains. Even a small income can be enough when it is used for what is truly needed. Even a large income can feel lacking when it is spent only on lifestyle.\u003c/p\u003e\n\u003cp\u003ePleasant words are not always true, and painful words are not always wrong.\u003c/p\u003e\n\u003cp\u003eLife is short. Let go of those who hurt you, and love those who truly care for you. 😊🤗\u003c/p\u003e\n\u003cp\u003eSource: Telegram Group\u003c/p\u003e\n", "language": "en" }, { "title" : "The Fish and the Turtle", "date_published" : "2020-07-09T06:03:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/ikan-dan-kura-kura/", "url" : "https://www.aantriono.com/en/ikan-dan-kura-kura/", "summary": " The Inspiring Story of the Fish and the Turtlefish in the sea Peace be upon you\u0026hellip; Below, Afra Media shares a short lesson through a dialogue between a fish and a turtle. I hope this is helpful.\nThe Dialogue Between Fish and TurtleA fish asked the turtle: \u0026ldquo;Why is it that whenever you have a problem, you always hide inside your shell?\u0026rdquo;\nThe turtle answered: \u0026ldquo;Is it really necessary for me to answer that question?\u0026rdquo;\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"the-inspiring-story-of-the-fish-and-the-turtle\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-inspiring-story-of-the-fish-and-the-turtle\" class=\"header-mark\" aria-label=\"Direct link to section: The Inspiring Story of the Fish and the Turtle\"\u003e\u003c/a\u003eThe Inspiring Story of the Fish and the Turtle\u003c/h2\u003e\u003cp\u003e\u003cfigure\u003e\u003ca class=\"lightgallery\" href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmLu7MJOL_NkALsy2pbJSMh8BqNDP3wZDMU-ejJNIIQgcf9qRVjgtBndbmWr1l1sQGp9XEXEzBIMba9Yf9qCjsGOMLLoBSVKwBPhPn_MusXO5b-7vdMj7LOA6hNzrTo6JsHhcdqsXDLuo/w200-h133/15820_ikan-dan-kura-kura.jpg\" title=\"marine animals\" data-thumbnail=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmLu7MJOL_NkALsy2pbJSMh8BqNDP3wZDMU-ejJNIIQgcf9qRVjgtBndbmWr1l1sQGp9XEXEzBIMba9Yf9qCjsGOMLLoBSVKwBPhPn_MusXO5b-7vdMj7LOA6hNzrTo6JsHhcdqsXDLuo/w200-h133/15820_ikan-dan-kura-kura.jpg\" data-sub-html=\"\u003ch2\u003efish in the sea\u003c/h2\u003e\u003cp\u003emarine animals\u003c/p\u003e\"\u003e\u003cimg loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmLu7MJOL_NkALsy2pbJSMh8BqNDP3wZDMU-ejJNIIQgcf9qRVjgtBndbmWr1l1sQGp9XEXEzBIMba9Yf9qCjsGOMLLoBSVKwBPhPn_MusXO5b-7vdMj7LOA6hNzrTo6JsHhcdqsXDLuo/w200-h133/15820_ikan-dan-kura-kura.jpg' alt=\"marine animals\" \u003e\u003c/a\u003e\u003cfigcaption class=\"image-caption\"\u003efish in the sea\u003c/figcaption\u003e\n\u003c/figure\u003e\u003c/p\u003e\n\u003cp\u003ePeace be upon you\u0026hellip; Below, Afra Media shares a short lesson through a dialogue between a fish and a turtle. I hope this is helpful.\u003c/p\u003e\n\u003ch2 id=\"the-dialogue-between-fish-and-turtle\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-dialogue-between-fish-and-turtle\" class=\"header-mark\" aria-label=\"Direct link to section: The Dialogue Between Fish and Turtle\"\u003e\u003c/a\u003eThe Dialogue Between Fish and Turtle\u003c/h2\u003e\u003cp\u003eA fish asked the turtle: \u0026ldquo;Why is it that whenever you have a problem, you always hide inside your shell?\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eThe turtle answered: \u0026ldquo;Is it really necessary for me to answer that question?\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eThe fish said: \u0026ldquo;Everyone in these waters wonders why you always withdraw into your shell whenever trouble comes.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eThe turtle replied: \u0026ldquo;Why should I be troubled by what others say? I am not avoiding anything. I am not running away from reality. I am simply seeking calm inside my shell.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eThe fish asked again: \u0026ldquo;But doesn\u0026rsquo;t it bother you to become the subject of conversation?\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eThe turtle answered: \u0026ldquo;That is one reason I live longer than many of you. You spend so much time watching my life that you forget to care for your own.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;In this life, each of us must make our own choices. Do your best, and let people say whatever they want.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Those who like you may still correct you when you are wrong. But those who dislike you may still blame you even when you are right.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;How much time is wasted worrying about other people\u0026rsquo;s lives, while forgetting your own need for rest, care, and reflection?\u0026rdquo;\u003c/p\u003e\n\u003cp\u003e\u0026ldquo;Love yourself by paying proper attention to your own responsibilities, because the one who is always busy with other people\u0026rsquo;s affairs will always feel lacking.\u0026rdquo;\u003c/p\u003e\n\u003cp\u003eMay we not be deceived by what people say. Be yourself. Be a good person, and keep doing good.\u003c/p\u003e\n\u003cp\u003eSource: Telegram Group\u003c/p\u003e", "language": "en" }, { "title" : "Algebra Form Material Questions", "date_published" : "2020-07-08T14:53:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/soal-materi-bentuk-aljabar/", "url" : "https://www.aantriono.com/en/soal-materi-bentuk-aljabar/", "summary": " Practice Questions on Algebraic Forms Below you have attached algebra material questions.\nplease do it properly.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"practice-questions-on-algebraic-forms\" class=\"headerLink\"\u003e\n \u003ca href=\"#practice-questions-on-algebraic-forms\" class=\"header-mark\" aria-label=\"Direct link to section: Practice Questions on Algebraic Forms\"\u003e\u003c/a\u003ePractice Questions on Algebraic Forms\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8XEeG063nBY5wKFdDzawBnGiuTzZVuj4E2pVtUCh-X7PtDlg0N3NjbPAGu-0Ym4fixQZzvTNaJg6U3XNapydfddU8bRwhmeQQGw6uUsUTn08fs2wSEBERkS1pRxC3V9I02KriEGU7Pm0/s1600/soal.png\" imageanchor=\"1\" style=\"clear: left; float: left; margin-bottom: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"300\" data-original-width=\"300\" height=\"200\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8XEeG063nBY5wKFdDzawBnGiuTzZVuj4E2pVtUCh-X7PtDlg0N3NjbPAGu-0Ym4fixQZzvTNaJg6U3XNapydfddU8bRwhmeQQGw6uUsUTn08fs2wSEBERkS1pRxC3V9I02KriEGU7Pm0/s200/soal.png\" width=\"200\" /\u003e\u003c/a\u003e\nBelow you have attached algebra material questions.\u003c/p\u003e\n\u003cp\u003eplease do it properly.\u003c/p\u003e\n\u003ch2 id=\"online-algebra-practice-form\" class=\"headerLink\"\u003e\n \u003ca href=\"#online-algebra-practice-form\" class=\"header-mark\" aria-label=\"Direct link to section: Online Algebra Practice Form\"\u003e\u003c/a\u003eOnline Algebra Practice Form\u003c/h2\u003e\u003ciframe allowfullscreen=\"\" frameborder=\"0\" height=\"480px\" marginheight=\"0\" marginwidth=\"0\" mozallowfullscreen=\"\" msallowfullscreen=\"\" src=\"https://forms.office.com/Pages/ResponsePage.aspx?id=IWuoD01i2EeN_yhVs5_MvgO2_k5oCTZFv98CPmYqJpRUNVlLVk80SVdZN1pHMEYwRTBDS1pNRjJaRi4u\u0026embed=true\" style=\"border: none; max-height: 100vh; max-width: 100%;\" webkitallowfullscreen=\"\" width=\"640px\"\u003e \u003c/iframe\u003e", "language": "en" }, { "title" : "Three Messages from Allah to Prophet Uzair", "date_published" : "2020-07-04T14:32:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/tiga-pesan-allah-kepada-nabi/", "url" : "https://www.aantriono.com/en/tiga-pesan-allah-kepada-nabi/", "summary": " Three Messages of Allah to Prophet Uzair\nIt is narrated that Allah once addressed the Prophet Uzair with three reminders. The message is simple but piercing: do not underestimate a sin merely because it seems small, do not belittle a blessing merely because it seems little, and do not turn to creation in complaint while forgetting the One who truly knows your condition.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"three-messages-of-allah-to-prophet-uzair\" class=\"headerLink\"\u003e\n \u003ca href=\"#three-messages-of-allah-to-prophet-uzair\" class=\"header-mark\" aria-label=\"Direct link to section: Three Messages of Allah to Prophet Uzair\"\u003e\u003c/a\u003eThree Messages of Allah to Prophet Uzair\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVVVseEB_Kmj9HQMO53FKi7cq6aqfxiF3bA5y56sWDgOTdIwrRHUeNeKAe2GxRG3SHCGEmN0FnLxZppt3hHJgz7JPmFWtUycrSeB2bMViSNWrx2K1p_YqUMv0vBJjC0A_8y5OqB-gDnfM/s200/pesan.png' alt=\"message\" \u003e\u003c/p\u003e\n\u003cp\u003eIt is narrated that Allah once addressed the Prophet Uzair with three reminders. The message is simple but piercing: do not underestimate a sin merely because it seems small, do not belittle a blessing merely because it seems little, and do not turn to creation in complaint while forgetting the One who truly knows your condition.\u003c/p\u003e\n\u003ch2 id=\"the-three-divine-advices\" class=\"headerLink\"\u003e\n \u003ca href=\"#the-three-divine-advices\" class=\"header-mark\" aria-label=\"Direct link to section: The Three Divine Advices\"\u003e\u003c/a\u003eThe Three Divine Advices\u003c/h2\u003e\u003col\u003e\n\u003cli\u003eIf you commit a small sin, then don\u0026rsquo;t look at the smallness of the sin, but look at the One whom you have disobeyed.\u003c/li\u003e\n\u003cli\u003eIf you receive a small good, then don\u0026rsquo;t look at the smallness of the goodness, but look at the One who has given you that blessing.\u003c/li\u003e\n\u003cli\u003eIf you are struck by a calamity, then do not complain about Me to My creatures, because I also never complain about you to My angels when you disobey Me.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eSource: \u003cem\u003eNashaihul Ibad\u003c/em\u003e — Imam Nawawi Al-Bantani\u003c/p\u003e", "language": "en" }, { "title" : "A Tongue That Never Dries", "date_published" : "2020-07-04T14:06:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/lisan-yang-tidak-kering/", "url" : "https://www.aantriono.com/en/lisan-yang-tidak-kering/", "summary": " Introduction\nIn the name of Allah\u0026hellip; Peace be upon you\u0026hellip;\nFriends, be grateful when your tongue is moved easily to pray and your heart no longer hesitates to ask Him. Not everyone finds it easy to turn to Allah, because many people are quicker to complain to others than to speak to Him. When your tongue becomes accustomed to asking from Him, that is a sign that you are being led toward His mercy and grace.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHtWIjA-1loX6wh-ntzYdMncV8Xl0JRLMy9mykcZPKo1QCntHdqc7Pl5SWoaK5UoLM5NMgZ7wqqmJ3A2-Kxw0k61RfoaZW_rg2HHL1aECNsSxicyvaNxgsVq95PSHZRzjliJcJfr7QhLQ/s200/JAGA-LISAN.png' alt=\"Guard your tongue\" \u003e\u003c/p\u003e\n\u003cp\u003eIn the name of Allah\u0026hellip; Peace be upon you\u0026hellip;\u003c/p\u003e\n\u003cp\u003eFriends, be grateful when your tongue is moved easily to pray and your heart no longer hesitates to ask Him. Not everyone finds it easy to turn to Allah, because many people are quicker to complain to others than to speak to Him. When your tongue becomes accustomed to asking from Him, that is a sign that you are being led toward His mercy and grace.\u003c/p\u003e\n\u003cp\u003eAsking from Him is not limited to formal worship. It should become a living attitude of the heart wherever you are. The kindness you receive from people should never distract you from the One who is the true source of all kindness. Pray wherever you are, because every sincere supplication brings the servant near to the Giver. Whoever often turns to Him will come to know that nearness more deeply.\u003c/p\u003e\n\u003ch2 id=\"when-your-tongue-is-moved-to-ask-it-means-he-wants-to-give-you\" class=\"headerLink\"\u003e\n \u003ca href=\"#when-your-tongue-is-moved-to-ask-it-means-he-wants-to-give-you\" class=\"header-mark\" aria-label=\"Direct link to section: When your tongue is moved to ask, it means He wants to give you.\"\u003e\u003c/a\u003eWhen your tongue is moved to ask, it means He wants to give you.\u003c/h2\u003e\u003cp\u003eSource: Al-Hikam — Strands of Wisdom from Ibn \u0026lsquo;Athaillah\u003c/p\u003e", "language": "en" }, { "title" : "Pinasti", "date_published" : "2020-07-04T13:41:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/pinasti/", "url" : "https://www.aantriono.com/en/pinasti/", "summary": " Reflection on Destiny and Effort In the name of Allah\u0026hellip; Peace be upon you\u0026hellip; At the moment, SAGUSABLOG is running an online workshop through Telegram chat, supported by learning modules and videos so participants can follow the lessons even while studying independently at home. Yet our plans do not always move in the direction we expect. Our task is simply to keep our focus on the goal and prepare ourselves as well as we can.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"reflection-on-destiny-and-effort\" class=\"headerLink\"\u003e\n \u003ca href=\"#reflection-on-destiny-and-effort\" class=\"header-mark\" aria-label=\"Direct link to section: Reflection on Destiny and Effort\"\u003e\u003c/a\u003eReflection on Destiny and Effort\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhETp6kj-qbbj5nwZYVxsu_OXHUMqTCa5gvbywNnp3kn3A3ybFD3rzpZz8TrTDxsaDysLFu4w2_dIaKaR5qs4OuuYdQA2DMwkthMwhuja5wpI8uhJ007d43zmvJUUasD_r1niIsgGjeNIc/s1600/takdir.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"takdir baik dan buruk\" border=\"0\" data-original-height=\"720\" data-original-width=\"1280\" height=\"180\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhETp6kj-qbbj5nwZYVxsu_OXHUMqTCa5gvbywNnp3kn3A3ybFD3rzpZz8TrTDxsaDysLFu4w2_dIaKaR5qs4OuuYdQA2DMwkthMwhuja5wpI8uhJ007d43zmvJUUasD_r1niIsgGjeNIc/w320-h180/takdir.jpg\" title=\"takdir baik dan buruk\" width=\"320\" /\u003e\u003c/a\u003e\nIn the name of Allah\u0026hellip;\nPeace be upon you\u0026hellip;\nAt the moment, SAGUSABLOG is running an online workshop through Telegram chat, supported by learning modules and videos so participants can follow the lessons even while studying independently at home. Yet our plans do not always move in the direction we expect. Our task is simply to keep our focus on the goal and prepare ourselves as well as we can.\u003c/p\u003e\n\u003ch2 id=\"accepting-his-decree\" class=\"headerLink\"\u003e\n \u003ca href=\"#accepting-his-decree\" class=\"header-mark\" aria-label=\"Direct link to section: Accepting His Decree\"\u003e\u003c/a\u003eAccepting His Decree\u003c/h2\u003e\u003cp\u003eAfter that, let His decree unfold. We never fully know what will happen once the arrow is released toward its target. Will it stay on course, or will it turn in another direction?\nGive yourself relief from the burden of trying to control everything. What has already been taken care of for you by the One above you does not need to be carried entirely by your own hands.\u003c/p\u003e\n\u003ch2 id=\"source-al-hikam-strands-of-wisdom-from-ibn-athaillah\" class=\"headerLink\"\u003e\n \u003ca href=\"#source-al-hikam-strands-of-wisdom-from-ibn-athaillah\" class=\"header-mark\" aria-label=\"Direct link to section: Source: Al-Hikam: Strands of Wisdom from Ibn \u0026lsquo;Athaillah\"\u003e\u003c/a\u003eSource: Al-Hikam: Strands of Wisdom from Ibn \u0026lsquo;Athaillah\u003c/h2\u003e", "language": "en" }, { "title" : "Sagusablog Activities", "date_published" : "2020-07-03T15:41:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/kegiatan-sagusablog/", "url" : "https://www.aantriono.com/en/kegiatan-sagusablog/", "summary": " Online Workshop for Educational BloggingSAGUSABLOG is currently hosting an online workshop through Telegram, supported by modules and learning videos so participants can follow the material more easily even while studying independently at home.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"online-workshop-for-educational-blogging\" class=\"headerLink\"\u003e\n \u003ca href=\"#online-workshop-for-educational-blogging\" class=\"header-mark\" aria-label=\"Direct link to section: Online Workshop for Educational Blogging\"\u003e\u003c/a\u003eOnline Workshop for Educational Blogging\u003c/h2\u003e\u003cp\u003eSAGUSABLOG is currently hosting an online workshop through Telegram, supported by modules and learning videos so participants can follow the material more easily even while studying independently at home.\u003c/p\u003e\n\u003ch2 id=\"profile-and-history-of-sagusablog\" class=\"headerLink\"\u003e\n \u003ca href=\"#profile-and-history-of-sagusablog\" class=\"header-mark\" aria-label=\"Direct link to section: Profile and History of SAGUSABLOG\"\u003e\u003c/a\u003eProfile and History of SAGUSABLOG\u003c/h2\u003e\u003cp\u003eSAGUSABLOG is one of the activity channels of the Indonesian Teachers Association. This channel provides knowledge and skills for creating blog-based learning media. To date, Sagusablog has reached its 43rd cohort and has trained thousands of teachers.\u003c/p\u003e\n\u003cp\u003e\u003cimg class=\"tw-inline\" loading=\"lazy\" src='https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVUbTzzfdO6bvFQjAD3dRbxGLLZgqNhjaEXAkVZNsltggGaW2iE3jz-FokBtyTIVhRDTIhI-JjTER-qX5hWoHvGLJyha01zIUEVO1-PS0j3dsUizpMySoDk-Ys4UQKjw36HSMIz6IajXM/w320-h320/logo-sagusablog-2018-2.png' alt=\"Sagusablog\" \u003e\u003c/p\u003e", "language": "en" }, { "title" : "MGMP Mathematics Middle School District. Central Lampung", "date_published" : "2020-07-02T16:12:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/mgmp-matematika-smp-kab-lampung-tengah/", "url" : "https://www.aantriono.com/en/mgmp-matematika-smp-kab-lampung-tengah/", "summary": " Implementation of CPD Training in MGMP Mathematics On October 9 2017 at SMP Kristen 2 Bandarjaya, Central Lampung Regency, face-to-face CPD training with APBN funds began for families A and E. 20 participants could attend 100%. They are very enthusiastic. Likewise with the Independent class consisting of 20 participants which started on October 11 2017 at the same place. The National Instructor in this activity is Gunanto, S.Pd. (Satya Dharma Sudjana Middle School) and Nur Hasanuri, M.Pd. (SMP N 1 Trimurjo).\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"implementation-of-cpd-training-in-mgmp-mathematics\" class=\"headerLink\"\u003e\n \u003ca href=\"#implementation-of-cpd-training-in-mgmp-mathematics\" class=\"header-mark\" aria-label=\"Direct link to section: Implementation of CPD Training in MGMP Mathematics\"\u003e\u003c/a\u003eImplementation of CPD Training in MGMP Mathematics\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9lgLyuI8NS-F9OthZDxIP7MBtpmZEWUPwhAzq0H-lbqs7ZJlL7ZSUqv7HWFVdj3cVrNkqRzorIaReJl1VIhaDb5tCp7YK_r0fQVB6PUP0IeSJqGzUO7hUS-B3TDRUtN_C827d-Jv8VA4/s1600/IMG_20171022_145827.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"kegiatan mgmp matematika lampung tengah\" border=\"0\" data-original-height=\"900\" data-original-width=\"1600\" height=\"180\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9lgLyuI8NS-F9OthZDxIP7MBtpmZEWUPwhAzq0H-lbqs7ZJlL7ZSUqv7HWFVdj3cVrNkqRzorIaReJl1VIhaDb5tCp7YK_r0fQVB6PUP0IeSJqGzUO7hUS-B3TDRUtN_C827d-Jv8VA4/w320-h180/IMG_20171022_145827.jpg\" title=\"kegiatan mgmp matematika\" width=\"320\" /\u003e\u003c/a\u003e\nOn October 9 2017 at SMP Kristen 2 Bandarjaya, Central Lampung Regency, face-to-face CPD training with APBN funds began for families A and E. 20 participants could attend 100%. They are very enthusiastic. Likewise with the Independent class consisting of 20 participants which started on October 11 2017 at the same place. The National Instructor in this activity is Gunanto, S.Pd. (Satya Dharma Sudjana Middle School) and Nur Hasanuri, M.Pd. (SMP N 1 Trimurjo).\u003c/p\u003e\n\u003ch2 id=\"opening-and-welcoming-speech\" class=\"headerLink\"\u003e\n \u003ca href=\"#opening-and-welcoming-speech\" class=\"header-mark\" aria-label=\"Direct link to section: Opening and Welcoming Speech\"\u003e\u003c/a\u003eOpening and Welcoming Speech\u003c/h2\u003e\u003cp\u003eMrs. Partilla Umar, S.Pd., M.\nM. as the person in charge of opening the training activities on behalf of the Head of the Central Lampung Regency Education Service said: \u0026ldquo;PKB training is a necessity for teachers. The results of the PKB training are expected to improve the quality of learning activities in schools. The Central Lampung Regency Education Office encourages teachers to continue to improve their competence, both using APBN funds and independently.\u0026rdquo;## In-On-In Training Model Structure\u003c/p\u003e\n\u003ch2 id=\"in-on-in-training-model-structure\" class=\"headerLink\"\u003e\n \u003ca href=\"#in-on-in-training-model-structure\" class=\"header-mark\" aria-label=\"Direct link to section: In-On-In Training Model Structure\"\u003e\u003c/a\u003eIn-On-In Training Model Structure\u003c/h2\u003e\u003cp\u003eActivities are carried out using the In-On-In 20-20-20 model. The first 20 JPs are in service learning activities, participants are guided by an instructor for 2 days, in 1 day participants discuss 1 KK so in 2 days they discuss 2 KKs, namely KK A and E. One KK consists of 2 modules, namely pedagogical and professional modules.\u003c/p\u003e", "language": "en" }, { "title" : "Grade 8 Mathematics Materials for the Odd Semester", "date_published" : "2020-07-02T14:05:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/materi-matematika-kelas-viii-semester/", "url" : "https://www.aantriono.com/en/materi-matematika-kelas-viii-semester/", "summary": " Mathematics Curriculum for Class VIII Odd SemesterClass VIII odd semester mathematics lesson material consists of 5 chapters which include:\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"mathematics-curriculum-for-class-viii-odd-semester\" class=\"headerLink\"\u003e\n \u003ca href=\"#mathematics-curriculum-for-class-viii-odd-semester\" class=\"header-mark\" aria-label=\"Direct link to section: Mathematics Curriculum for Class VIII Odd Semester\"\u003e\u003c/a\u003eMathematics Curriculum for Class VIII Odd Semester\u003c/h2\u003e\u003cp\u003e\u003cimg alt=\"buku matematika kelas 8 smp\" border=\"0\" data-original-height=\"451\" data-original-width=\"786\" height=\"183\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiszzsCpPzDHq-X_k3Q1AR-a1gGPZz4h2tkPCCR00W3PoTdMn91YFB7_wkDAwoPs0nBgrgWPVptFLRDSRGYNuZ8owH1sOd8AXd6aTS9N9Fy1ZZ6j_ST54d-Cs-cxjGfi91gRHa8kICl2u8/w320-h183/gambar+bs+mtk+8.jpg\" title=\"buku matematika kelas 8\" width=\"320\" /\u003eClass VIII odd semester mathematics lesson material consists of 5 chapters which include:\u003c/p\u003e\n\u003ch2 id=\"summary-of-topics-by-chapter\" class=\"headerLink\"\u003e\n \u003ca href=\"#summary-of-topics-by-chapter\" class=\"header-mark\" aria-label=\"Direct link to section: Summary of Topics by Chapter\"\u003e\u003c/a\u003eSummary of Topics by Chapter\u003c/h2\u003e\u003cp\u003eChapter 1 Number Patterns1.1 Determining the Equation of a Number Series1.2 Determining the Similarities of an Object Configuration\u003c/p\u003e\n\u003cp\u003eChapter 2 Cartesian Coordinates2.1 Point Position Regarding the X-Axis and Y-Axis2.2 Position of Points Regarding the Origin (0, 0) and Specific Points (a, b)2.3 Line Position Regarding the X-Axis and Y-Axis\u003c/p\u003e\n\u003cp\u003eChapter 3 Relations and Functions3.1 Understanding the Form of Relational Presentation3.2 Understanding Function Characteristics3.3 Understanding Function Presentation Forms3.4 Understanding One-on-One Correspondence\u003c/p\u003e\n\u003cp\u003eChapter 4 Straight Line Equations4.1 Graph of Straight Line Equations4.2 Determining the Slope of a Straight Line Equation4.3 Equation Form of a Straight Line with Slope m and Through Point (x1, y1)4.4 Properties of Straight Line Equations\u003c/p\u003e\n\u003cp\u003eChapter 5 Systems of Linear Equations in Two Variables5.1 Understand the concept of linear equations in two variables5.2 Solving Systems of Linear Equations in Two Variables by Drawing Graphs5.3 Solving Systems of Linear Equations in Two Variables by Substitution5.4 Solving Systems of Linear Equations in Two Variables by Elimination5.5 Solving Systems of Linear Equations in Two Special Variables\u003c/p\u003e", "language": "en" }, { "title" : "Class 8 Mathematics PAT Questions for the 2019/2020 Academic Year", "date_published" : "2020-07-02T07:16:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/soal-pat-matematika-kelas-8-tahun/", "url" : "https://www.aantriono.com/en/soal-pat-matematika-kelas-8-tahun/", "summary": " End of Year Assessment (PAT) Mathematics Class 8 Peace be upon you\u0026hellip; Good morning, Grade 8 students. I hope you are all well. Please do the following questions correctly!\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"end-of-year-assessment-pat-mathematics-class-8\" class=\"headerLink\"\u003e\n \u003ca href=\"#end-of-year-assessment-pat-mathematics-class-8\" class=\"header-mark\" aria-label=\"Direct link to section: End of Year Assessment (PAT) Mathematics Class 8\"\u003e\u003c/a\u003eEnd of Year Assessment (PAT) Mathematics Class 8\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc-lRyBWWsBwkkYohqxAW1T50OZEFSG--F6g9j7Ir6KrgcbVmK--uUUrhOk3JPjINWb3DkHClkHDRh_erF9kDQBVkVUr_HjiY49nj2BJq6-dvaBjIsGUp4skW8_4wOF-A0Fk335-1w6_B7UcF_w_z86tgKrbbSJLtpLkaoO9M6LOmiqGbn3_7GFeUJ/s1280/soal.png\" imageanchor=\"1\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg alt=\"soal penilaian akhir tahun kelas 8\" border=\"0\" data-original-height=\"400\" data-original-width=\"1280\" height=\"125\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc-lRyBWWsBwkkYohqxAW1T50OZEFSG--F6g9j7Ir6KrgcbVmK--uUUrhOk3JPjINWb3DkHClkHDRh_erF9kDQBVkVUr_HjiY49nj2BJq6-dvaBjIsGUp4skW8_4wOF-A0Fk335-1w6_B7UcF_w_z86tgKrbbSJLtpLkaoO9M6LOmiqGbn3_7GFeUJ/w400-h125/soal.png\" title=\"soal penilaian akhir tahun\" width=\"400\" /\u003e\u003c/a\u003e\nPeace be upon you\u0026hellip; Good morning, Grade 8 students. I hope you are all well.\nPlease do the following questions correctly!\u003c/p\u003e\n\u003cp\u003eThe token is 12345\nGood luck with your work, good luck.\u003c/p\u003e\n\u003ch2 id=\"online-pat-mathematics-assessment-form\" class=\"headerLink\"\u003e\n \u003ca href=\"#online-pat-mathematics-assessment-form\" class=\"header-mark\" aria-label=\"Direct link to section: Online PAT Mathematics Assessment Form\"\u003e\u003c/a\u003eOnline PAT Mathematics Assessment Form\u003c/h2\u003e\u003cblockquote\u003e\n \u003cp\u003eThe online assessment form is no longer available from the service provider (the Google Form was removed after the 2019/2020 academic year exam period ended).\u003c/p\u003e\n\u003c/blockquote\u003e", "language": "en" }, { "title" : "Sagusablog", "date_published" : "2020-07-02T06:54:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/sagusablog/", "url" : "https://www.aantriono.com/en/sagusablog/", "summary": " Understanding the IGI SAGUSABLOG ProgramSAGUSABLOG or also called One Teacher One Blog is one of the activities initiated** IGI **(Indonesian Teachers Association). **SAGUSABLOG\n** was first carried out in Surabaya in a series of IT-Based Productive Literacy TOT activities in the Mobile Blogging class with trainer Amin Mungamar /Mr. Mung, and Dahli Ahmad/Kang Dahli. ## Online Workshop and Documentation\nOnline Workshop and DocumentationCurrently, SAGUSABLOG is opening an online workshop using chat facilities via the Telegram application, assisted by learning modules and videos so that it is easy for online workshop participants to understand even though they are studying independently at home. Below I have included a video of the Sagusablog class 41 activities.\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"understanding-the-igi-sagusablog-program\" class=\"headerLink\"\u003e\n \u003ca href=\"#understanding-the-igi-sagusablog-program\" class=\"header-mark\" aria-label=\"Direct link to section: Understanding the IGI SAGUSABLOG Program\"\u003e\u003c/a\u003eUnderstanding the IGI SAGUSABLOG Program\u003c/h2\u003e\u003cp\u003eSAGUSABLOG or also called One Teacher One Blog is one of the activities initiated**\u003ca href=\"https://www.igi.or.id/\"\u003e IGI \u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e**(Indonesian Teachers Association). **\u003ca href=\"http://www.sagusablog.com/\"\u003eSAGUSABLOG\u003c/a\u003e\u003c/p\u003e\n\u003cp\u003e** was first carried out in Surabaya in a series of IT-Based Productive Literacy TOT activities in the Mobile Blogging class with trainer Amin Mungamar /Mr. Mung, and Dahli Ahmad/Kang Dahli. ## Online Workshop and Documentation\u003c/p\u003e\n\u003ch2 id=\"online-workshop-and-documentation\" class=\"headerLink\"\u003e\n \u003ca href=\"#online-workshop-and-documentation\" class=\"header-mark\" aria-label=\"Direct link to section: Online Workshop and Documentation\"\u003e\u003c/a\u003eOnline Workshop and Documentation\u003c/h2\u003e\u003cp\u003eCurrently, SAGUSABLOG is opening an online workshop using chat facilities via the Telegram application, assisted by learning modules and videos so that it is easy for online workshop participants to understand even though they are studying independently at home. Below I have included a video of the Sagusablog class 41 activities.\u003c/p\u003e\n\u003c!--more--\u003e\n\u003ciframe allowfullscreen=\"\" class=\"YOUTUBE-iframe-video\" data-thumbnail-src=\"https://i.ytimg.com/vi/hxqdoHk6T8A/0.jpg\" frameborder=\"0\" height=\"266\" src=\"https://www.youtube.com/embed/hxqdoHk6T8A?feature=player_embedded\" width=\"320\"\u003e\u003c/iframe\u003e", "language": "en" }, { "title" : "Lesson plan", "date_published" : "2020-07-01T07:14:00Z", "date_modified" : "2026-08-08T23:35:13Z", "id" : "https://www.aantriono.com/en/rencana-pelaksanaan-pembelajaran/", "url" : "https://www.aantriono.com/en/rencana-pelaksanaan-pembelajaran/", "summary": " Introduction\nUnderstanding RPPA learning implementation plan (RPP) is a plan that describes the procedures and organization of learning to achieve a basic competency set out in the Content Standards and described in the syllabus (Kunandar, 2011: 263).\n", "author" : { "name" : "Aan Triono" }, "content_html" : "\u003ch2 id=\"introduction\" class=\"headerLink\"\u003e\n \u003ca href=\"#introduction\" class=\"header-mark\" aria-label=\"Direct link to section: Introduction\"\u003e\u003c/a\u003eIntroduction\u003c/h2\u003e\u003cp\u003e\u003ca href=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjzDfokfzDmNP9u6wVNr0UPWNnX5PxWyiyM9yvUwl7GX4OMPHrauh6Pr8ZMj3htiQOpa-sRkUXiSE67Q64Ck2gIHGNQu-nxwLJj_2g-TqIlEmcNHVIK9Bfr3Ybb9luKH8pI9FOdzKqF2I/s1600/rpp.jpg\" style=\"margin-left: 1em; margin-right: 1em;\"\u003e\u003cimg border=\"0\" data-original-height=\"181\" data-original-width=\"278\" src=\"https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjzDfokfzDmNP9u6wVNr0UPWNnX5PxWyiyM9yvUwl7GX4OMPHrauh6Pr8ZMj3htiQOpa-sRkUXiSE67Q64Ck2gIHGNQu-nxwLJj_2g-TqIlEmcNHVIK9Bfr3Ybb9luKH8pI9FOdzKqF2I/s16000/rpp.jpg\" /\u003e\u003c/a\u003e\u003c/p\u003e\n\u003ch2 id=\"understanding-rpp\" class=\"headerLink\"\u003e\n \u003ca href=\"#understanding-rpp\" class=\"header-mark\" aria-label=\"Direct link to section: Understanding RPP\"\u003e\u003c/a\u003eUnderstanding RPP\u003c/h2\u003e\u003cp\u003eA learning implementation plan (RPP) is a plan that describes the procedures and organization of learning to achieve a basic competency set out in the Content Standards and described in the syllabus (Kunandar, 2011: 263).\u003c/p\u003e\n\u003cp\u003eBased on Republic of Indonesia Government Regulation Number 19 of 2005 Article 20, it is stated that \u0026ldquo;Learning process planning includes a syllabus and learning plan which contains at least learning objectives, learning resources and assessment of learning outcomes.\u0026rdquo;\u003c/p\u003e\n\u003ch2 id=\"purpose-and-function-of-rpp\" class=\"headerLink\"\u003e\n \u003ca href=\"#purpose-and-function-of-rpp\" class=\"header-mark\" aria-label=\"Direct link to section: Purpose and Function of RPP\"\u003e\u003c/a\u003ePurpose and Function of RPP\u003c/h2\u003e\u003cp\u003eThe objectives of the learning implementation plan are to: (1) simplify, expedite and improve the results of the teaching and learning process; (2) by preparing learning plans professionally, systematically and effectively, teachers will be able to see, observe, analyze and predict learning programs as a logical and planned framework (Kunandar, 2011: 264).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eLearning plan function\u003c/strong\u003e is as a reference for teachers to carry out teaching and learning activities (learning activities) so that they are more focused and run effectively and efficiently (Kunandar, 2011: 264).\u003c/p\u003e\n\u003ch2 id=\"elements-that-need-to-be-considered-when-preparing-a-rpp\" class=\"headerLink\"\u003e\n \u003ca href=\"#elements-that-need-to-be-considered-when-preparing-a-rpp\" class=\"header-mark\" aria-label=\"Direct link to section: Elements that need to be considered when preparing a RPP\"\u003e\u003c/a\u003eElements that need to be considered when preparing a RPP\u003c/h2\u003e\u003col\u003e\n\u003cli\u003erefers to the basic competencies and abilities that students must master, as well as learning materials and sub-materials, learning experiences that have been developed in the syllabus;\u003c/li\u003e\n\u003cli\u003euse various approaches that are appropriate to the material that provides life skills according to everyday problems and environments;\u003c/li\u003e\n\u003cli\u003eusing appropriate methods and media, which bring students closer to direct experience;\u003c/li\u003e\n\u003cli\u003eassessment with a comprehensive and continuous testing system is based on a testing system developed in line with syllabus development (Kunandar, 2011: 265).\u003c/li\u003e\n\u003c/ol\u003e\nBelow I have included the lesson plan in 1 sheet format\n\u003ciframe height=\"480\" src=\"https://drive.google.com/file/d/1jVB9oJgPpXFvLOoMy9HwN8tiVcUVDCWU/preview\" width=\"600\"\u003e\u003c/iframe\u003e\n", "language": "en" }, ] }